| L(s) = 1 | + 5.90e4·3-s + 3.10e6·5-s − 3.63e8·7-s + 3.48e9·9-s − 1.45e10·11-s + 1.13e11·13-s + 1.83e11·15-s − 8.58e12·17-s + 2.92e13·19-s − 2.14e13·21-s + 1.55e14·23-s − 4.67e14·25-s + 2.05e14·27-s + 2.40e15·29-s − 2.23e15·31-s − 8.61e14·33-s − 1.12e15·35-s − 3.07e16·37-s + 6.69e15·39-s − 1.03e17·41-s + 1.65e17·43-s + 1.08e16·45-s + 6.65e16·47-s − 4.26e17·49-s − 5.07e17·51-s + 4.35e17·53-s − 4.53e16·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.142·5-s − 0.486·7-s + 1/3·9-s − 0.169·11-s + 0.228·13-s + 0.0822·15-s − 1.03·17-s + 1.09·19-s − 0.280·21-s + 0.784·23-s − 0.979·25-s + 0.192·27-s + 1.05·29-s − 0.490·31-s − 0.0978·33-s − 0.0692·35-s − 1.05·37-s + 0.131·39-s − 1.20·41-s + 1.16·43-s + 0.0474·45-s + 0.184·47-s − 0.763·49-s − 0.596·51-s + 0.341·53-s − 0.0241·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{10} T \) |
| good | 5 | \( 1 - 124398 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 + 51900560 p T + p^{21} T^{2} \) |
| 11 | \( 1 + 1325621196 p T + p^{21} T^{2} \) |
| 13 | \( 1 - 113350790702 T + p^{21} T^{2} \) |
| 17 | \( 1 + 505258211646 p T + p^{21} T^{2} \) |
| 19 | \( 1 - 1536996803884 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 155899214954280 T + p^{21} T^{2} \) |
| 29 | \( 1 - 2400788707090758 T + p^{21} T^{2} \) |
| 31 | \( 1 + 2239820676947000 T + p^{21} T^{2} \) |
| 37 | \( 1 + 30785069383298890 T + p^{21} T^{2} \) |
| 41 | \( 1 + 103207571041281030 T + p^{21} T^{2} \) |
| 43 | \( 1 - 165557270617488124 T + p^{21} T^{2} \) |
| 47 | \( 1 - 66587216226477408 T + p^{21} T^{2} \) |
| 53 | \( 1 - 435422766592881630 T + p^{21} T^{2} \) |
| 59 | \( 1 + 5534365798259081316 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7176205164722961202 T + p^{21} T^{2} \) |
| 67 | \( 1 - 15755449453068299812 T + p^{21} T^{2} \) |
| 71 | \( 1 + 26457854874259376232 T + p^{21} T^{2} \) |
| 73 | \( 1 - 13471249335464801450 T + p^{21} T^{2} \) |
| 79 | \( 1 - 16886125085525986840 T + p^{21} T^{2} \) |
| 83 | \( 1 - \)\(17\!\cdots\!72\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 + \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(94\!\cdots\!18\)\( T + p^{21} T^{2} \) |
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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
−10.80480510966912493902426492851, −9.642977187219498813793928704757, −8.743433005730774181418152985782, −7.48665338453898285975471538348, −6.38355298453282589869415340167, −4.99294708078339142550374016889, −3.64770574382722130777440088937, −2.64015138473391727632372501585, −1.40149024768447915211220761114, 0,
1.40149024768447915211220761114, 2.64015138473391727632372501585, 3.64770574382722130777440088937, 4.99294708078339142550374016889, 6.38355298453282589869415340167, 7.48665338453898285975471538348, 8.743433005730774181418152985782, 9.642977187219498813793928704757, 10.80480510966912493902426492851
