| L(s) = 1 | + 5.03·2-s − 8.47·3-s + 17.3·4-s − 42.6·6-s − 3.81·7-s + 46.9·8-s + 44.8·9-s − 52.3·11-s − 146.·12-s + 8.06·13-s − 19.2·14-s + 97.5·16-s + 17·17-s + 225.·18-s − 66.5·19-s + 32.3·21-s − 263.·22-s − 180.·23-s − 397.·24-s + 40.5·26-s − 151.·27-s − 66.1·28-s − 41.2·29-s − 34.9·31-s + 115.·32-s + 443.·33-s + 85.5·34-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 1.63·3-s + 2.16·4-s − 2.90·6-s − 0.206·7-s + 2.07·8-s + 1.66·9-s − 1.43·11-s − 3.53·12-s + 0.171·13-s − 0.366·14-s + 1.52·16-s + 0.242·17-s + 2.95·18-s − 0.803·19-s + 0.336·21-s − 2.55·22-s − 1.63·23-s − 3.38·24-s + 0.305·26-s − 1.07·27-s − 0.446·28-s − 0.264·29-s − 0.202·31-s + 0.638·32-s + 2.34·33-s + 0.431·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 5.03T + 8T^{2} \) |
| 3 | \( 1 + 8.47T + 27T^{2} \) |
| 7 | \( 1 + 3.81T + 343T^{2} \) |
| 11 | \( 1 + 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.06T + 2.19e3T^{2} \) |
| 19 | \( 1 + 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 15.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
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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
−10.65582148305419295010878568267, −10.11571479183306067922846111794, −8.021778024239753841482710190722, −6.85672979542525907518217516971, −6.11544185169223641764041228683, −5.39660698371921280187530146822, −4.73122437496075906569633285957, −3.60327630568298560647343117197, −2.08331760827142492881258176376, 0,
2.08331760827142492881258176376, 3.60327630568298560647343117197, 4.73122437496075906569633285957, 5.39660698371921280187530146822, 6.11544185169223641764041228683, 6.85672979542525907518217516971, 8.021778024239753841482710190722, 10.11571479183306067922846111794, 10.65582148305419295010878568267
