| L(s) = 1 | + 952·3-s − 5.36e4·5-s + 3.33e5·7-s − 1.16e8·9-s − 4.30e8·11-s + 2.66e9·13-s − 5.10e7·15-s + 6.06e10·17-s − 1.78e11·19-s + 3.17e8·21-s − 5.28e11·23-s − 1.01e12·25-s − 1.00e11·27-s − 7.24e12·29-s + 1.87e12·31-s − 4.10e11·33-s − 1.78e10·35-s − 2.03e13·37-s + 2.53e12·39-s − 1.76e12·41-s − 1.93e14·43-s + 6.27e12·45-s − 1.00e14·47-s − 3.30e14·49-s + 5.77e13·51-s − 3.17e14·53-s + 2.31e13·55-s + ⋯ |
| L(s) = 1 | + 0.0837·3-s − 0.0613·5-s + 0.0218·7-s − 0.905·9-s − 0.606·11-s + 0.906·13-s − 0.00514·15-s + 2.10·17-s − 2.41·19-s + 0.00182·21-s − 1.40·23-s − 1.33·25-s − 0.0685·27-s − 2.68·29-s + 0.395·31-s − 0.0507·33-s − 0.00134·35-s − 0.951·37-s + 0.0759·39-s − 0.0344·41-s − 2.52·43-s + 0.0555·45-s − 0.617·47-s − 1.42·49-s + 0.176·51-s − 0.701·53-s + 0.0372·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 952 T + 13096294 p^{2} T^{2} - 952 p^{17} T^{3} + p^{34} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 10724 p T + 8162342086 p^{3} T^{2} + 10724 p^{18} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 333168 T + 6750380073134 p^{2} T^{2} - 333168 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 430974680 T + 24621881623740466 p T^{2} + 430974680 p^{17} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 205193996 p T + 22476549286509102 p^{2} T^{2} - 205193996 p^{18} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3569029604 p T + \)\(22\!\cdots\!74\)\( T^{2} - 3569029604 p^{18} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 178629960040 T + \)\(18\!\cdots\!42\)\( T^{2} + 178629960040 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 528756594608 T + \)\(30\!\cdots\!38\)\( T^{2} + 528756594608 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7240660091460 T + \)\(26\!\cdots\!54\)\( T^{2} + 7240660091460 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1878351140288 T + \)\(42\!\cdots\!42\)\( T^{2} - 1878351140288 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 20332464566580 T + \)\(11\!\cdots\!98\)\( T^{2} + 20332464566580 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1763041905324 T + \)\(20\!\cdots\!10\)\( T^{2} + 1763041905324 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 193394525968664 T + \)\(20\!\cdots\!54\)\( T^{2} + 193394525968664 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 100763837765472 T + \)\(33\!\cdots\!70\)\( T^{2} + 100763837765472 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 317818146060052 T + \)\(20\!\cdots\!02\)\( T^{2} + 317818146060052 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1262050788321736 T + \)\(18\!\cdots\!06\)\( T^{2} - 1262050788321736 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2765859692723708 T + \)\(50\!\cdots\!62\)\( T^{2} - 2765859692723708 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1963006544550088 T + \)\(22\!\cdots\!46\)\( T^{2} + 1963006544550088 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 483639107104528 T + \)\(58\!\cdots\!42\)\( T^{2} + 483639107104528 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2176892348591212 T + \)\(18\!\cdots\!58\)\( T^{2} + 2176892348591212 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5295839905627744 T + \)\(33\!\cdots\!98\)\( T^{2} - 5295839905627744 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9972518018887144 T + \)\(81\!\cdots\!34\)\( T^{2} + 9972518018887144 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 711099036813900 T + \)\(23\!\cdots\!82\)\( T^{2} + 711099036813900 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 114870546609971908 T + \)\(15\!\cdots\!90\)\( T^{2} - 114870546609971908 p^{17} T^{3} + p^{34} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.57461372466646405761651201636, −14.14581907229050531524887911117, −13.08221432243669304829056285531, −12.87735885974373990996417039248, −11.60042591869005389231584157134, −11.49646696807449282711724481392, −10.27195875042109830554273840852, −10.00072563090849223550820115262, −8.796882922356220580266623693545, −8.164551967752457317837648874472, −7.73439384461476505375235945870, −6.40895531718061592641317140630, −5.83614827633054183024927021022, −5.13716553159828243183061773678, −3.77212879853955668251437041419, −3.47172964460832010038120717437, −2.14672440721933790658318161356, −1.61354746377876310424232025531, 0, 0,
1.61354746377876310424232025531, 2.14672440721933790658318161356, 3.47172964460832010038120717437, 3.77212879853955668251437041419, 5.13716553159828243183061773678, 5.83614827633054183024927021022, 6.40895531718061592641317140630, 7.73439384461476505375235945870, 8.164551967752457317837648874472, 8.796882922356220580266623693545, 10.00072563090849223550820115262, 10.27195875042109830554273840852, 11.49646696807449282711724481392, 11.60042591869005389231584157134, 12.87735885974373990996417039248, 13.08221432243669304829056285531, 14.14581907229050531524887911117, 14.57461372466646405761651201636
