| L(s) = 1 | + 8.49·3-s − 3.12·5-s + 10.0·7-s + 45.1·9-s + 14.6·11-s + 35.8·13-s − 26.5·15-s + 22.4·17-s − 122.·19-s + 85.2·21-s + 19.7·23-s − 115.·25-s + 154.·27-s + 165.·29-s + 31·31-s + 124.·33-s − 31.3·35-s + 160.·37-s + 304.·39-s − 373.·41-s − 271.·43-s − 141.·45-s − 30.7·47-s − 242.·49-s + 190.·51-s − 300.·53-s − 45.6·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.279·5-s + 0.541·7-s + 1.67·9-s + 0.400·11-s + 0.765·13-s − 0.457·15-s + 0.320·17-s − 1.47·19-s + 0.885·21-s + 0.179·23-s − 0.921·25-s + 1.10·27-s + 1.06·29-s + 0.179·31-s + 0.654·33-s − 0.151·35-s + 0.713·37-s + 1.25·39-s − 1.42·41-s − 0.962·43-s − 0.468·45-s − 0.0954·47-s − 0.706·49-s + 0.524·51-s − 0.777·53-s − 0.112·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.744624207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.744624207\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - 31T \) |
| good | 3 | \( 1 - 8.49T + 27T^{2} \) |
| 5 | \( 1 + 3.12T + 125T^{2} \) |
| 7 | \( 1 - 10.0T + 343T^{2} \) |
| 11 | \( 1 - 14.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 373.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 30.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 1.16T + 2.05e5T^{2} \) |
| 61 | \( 1 + 864.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 337.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 292.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 727.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 151.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 224.T + 9.12e5T^{2} \) |
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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
−13.25624668509001670428682899829, −12.02818956935115327501959368091, −10.73433604018597956685568708004, −9.524045527972126030898480445112, −8.459375425529051171324837145235, −7.966059847443717248673514375577, −6.50696232992217486231764217639, −4.45697952306124053103247088075, −3.31798692341547206175071377900, −1.78623670175239427188982230724,
1.78623670175239427188982230724, 3.31798692341547206175071377900, 4.45697952306124053103247088075, 6.50696232992217486231764217639, 7.966059847443717248673514375577, 8.459375425529051171324837145235, 9.524045527972126030898480445112, 10.73433604018597956685568708004, 12.02818956935115327501959368091, 13.25624668509001670428682899829
