| L(s) = 1 | + 7-s − 3·9-s − 11-s + 6·13-s − 2·19-s − 4·23-s − 5·25-s + 2·29-s − 2·31-s − 2·37-s − 8·41-s + 2·47-s + 49-s + 10·53-s − 4·59-s − 10·61-s − 3·63-s + 4·67-s − 8·73-s − 77-s − 8·79-s + 9·81-s − 2·83-s − 6·89-s + 6·91-s + 2·97-s + 3·99-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.301·11-s + 1.66·13-s − 0.458·19-s − 0.834·23-s − 25-s + 0.371·29-s − 0.359·31-s − 0.328·37-s − 1.24·41-s + 0.291·47-s + 1/7·49-s + 1.37·53-s − 0.520·59-s − 1.28·61-s − 0.377·63-s + 0.488·67-s − 0.936·73-s − 0.113·77-s − 0.900·79-s + 81-s − 0.219·83-s − 0.635·89-s + 0.628·91-s + 0.203·97-s + 0.301·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.082829361255205378857056014456, −7.25098837756828571927486975668, −6.17489255432857712626897865109, −5.90295886078658282964934667135, −5.03028859045657826817868814043, −4.03247490116728845293852440262, −3.40216562410863542544669133877, −2.36702317337257336852464968661, −1.42016359685483307745209014993, 0,
1.42016359685483307745209014993, 2.36702317337257336852464968661, 3.40216562410863542544669133877, 4.03247490116728845293852440262, 5.03028859045657826817868814043, 5.90295886078658282964934667135, 6.17489255432857712626897865109, 7.25098837756828571927486975668, 8.082829361255205378857056014456
