| L(s) = 1 | + 2.69·3-s + 2.06·5-s − 7-s + 4.28·9-s − 6.03·11-s − 2.94·13-s + 5.57·15-s + 17-s − 6.57·19-s − 2.69·21-s + 6.75·23-s − 0.728·25-s + 3.45·27-s + 1.85·29-s − 10.5·31-s − 16.2·33-s − 2.06·35-s + 4.17·37-s − 7.94·39-s − 8.24·41-s − 7.78·43-s + 8.84·45-s − 4.43·47-s + 49-s + 2.69·51-s − 2.60·53-s − 12.4·55-s + ⋯ |
| L(s) = 1 | + 1.55·3-s + 0.924·5-s − 0.377·7-s + 1.42·9-s − 1.81·11-s − 0.816·13-s + 1.43·15-s + 0.242·17-s − 1.50·19-s − 0.588·21-s + 1.40·23-s − 0.145·25-s + 0.664·27-s + 0.344·29-s − 1.89·31-s − 2.83·33-s − 0.349·35-s + 0.685·37-s − 1.27·39-s − 1.28·41-s − 1.18·43-s + 1.31·45-s − 0.647·47-s + 0.142·49-s + 0.377·51-s − 0.357·53-s − 1.68·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 11 | \( 1 + 6.03T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 4.17T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 + 6.62T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + 7.47T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.69216369032715382163437645332, −7.01534403557589467241374535668, −6.22748569535584254385353104707, −5.24898990059356384808187878648, −4.80768519560051931933219823037, −3.61300884634249185435004296941, −2.96192669349876259734167893998, −2.31491246289632490673065205884, −1.80362041593907853443372415476, 0,
1.80362041593907853443372415476, 2.31491246289632490673065205884, 2.96192669349876259734167893998, 3.61300884634249185435004296941, 4.80768519560051931933219823037, 5.24898990059356384808187878648, 6.22748569535584254385353104707, 7.01534403557589467241374535668, 7.69216369032715382163437645332
