| L(s) = 1 | − 9.04·3-s − 2.15·5-s + 27.7·7-s + 54.7·9-s + 0.351·11-s + 65.3·13-s + 19.5·15-s − 68.3·17-s − 89.8·19-s − 251.·21-s + 110.·23-s − 120.·25-s − 251.·27-s + 29·29-s − 258.·31-s − 3.17·33-s − 59.9·35-s + 128.·37-s − 591.·39-s + 283.·41-s − 339.·43-s − 118.·45-s − 147.·47-s + 429.·49-s + 617.·51-s − 518.·53-s − 0.758·55-s + ⋯ |
| L(s) = 1 | − 1.74·3-s − 0.192·5-s + 1.50·7-s + 2.02·9-s + 0.00963·11-s + 1.39·13-s + 0.335·15-s − 0.974·17-s − 1.08·19-s − 2.61·21-s + 1.00·23-s − 0.962·25-s − 1.79·27-s + 0.185·29-s − 1.49·31-s − 0.0167·33-s − 0.289·35-s + 0.569·37-s − 2.42·39-s + 1.07·41-s − 1.20·43-s − 0.391·45-s − 0.457·47-s + 1.25·49-s + 1.69·51-s − 1.34·53-s − 0.00185·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 9.04T + 27T^{2} \) |
| 5 | \( 1 + 2.15T + 125T^{2} \) |
| 7 | \( 1 - 27.7T + 343T^{2} \) |
| 11 | \( 1 - 0.351T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 68.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 110.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 283.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 518.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 102.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 791.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 287.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 546.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 260.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 949.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.424392531478428416496434244411, −7.62054559699169812468325615316, −6.66976815931012923440983400960, −6.06219818410317185581475606634, −5.23163533675232470460631608890, −4.57289847072233145822261584332, −3.85631844390744526384646516353, −1.95371764650103523748548983196, −1.15143570614474843517151662041, 0,
1.15143570614474843517151662041, 1.95371764650103523748548983196, 3.85631844390744526384646516353, 4.57289847072233145822261584332, 5.23163533675232470460631608890, 6.06219818410317185581475606634, 6.66976815931012923440983400960, 7.62054559699169812468325615316, 8.424392531478428416496434244411
