| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 5·17-s + 3·19-s + 20-s − 22-s − 4·23-s + 25-s + 26-s − 28-s + 9·29-s − 11·31-s + 32-s − 5·34-s − 35-s + 7·37-s + 3·38-s + 40-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 0.688·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.67·29-s − 1.97·31-s + 0.176·32-s − 0.857·34-s − 0.169·35-s + 1.15·37-s + 0.486·38-s + 0.158·40-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.23808093862226, −16.14597720899613, −15.33063992985671, −14.89161262521672, −14.19448693058453, −13.58179126038017, −13.36534797140962, −12.63511700334423, −12.17496906561506, −11.42190074209811, −10.92297330553926, −10.32327742785698, −9.648775517137769, −9.124044389594783, −8.316204302714942, −7.709614508396101, −6.891853538433503, −6.369075794374553, −5.865000529155674, −5.028478766579931, −4.540074150587092, −3.644577170106131, −2.999000014456197, −2.212312109932931, −1.412621677839956, 0,
1.412621677839956, 2.212312109932931, 2.999000014456197, 3.644577170106131, 4.540074150587092, 5.028478766579931, 5.865000529155674, 6.369075794374553, 6.891853538433503, 7.709614508396101, 8.316204302714942, 9.124044389594783, 9.648775517137769, 10.32327742785698, 10.92297330553926, 11.42190074209811, 12.17496906561506, 12.63511700334423, 13.36534797140962, 13.58179126038017, 14.19448693058453, 14.89161262521672, 15.33063992985671, 16.14597720899613, 16.23808093862226
