| L(s) = 1 | + 3.60·2-s + 4.96·4-s + 15.9·5-s − 31.7·7-s − 10.9·8-s + 57.3·10-s + 41.2·13-s − 114.·14-s − 79.0·16-s + 21.4·17-s − 135.·19-s + 79.1·20-s − 4.92·23-s + 128.·25-s + 148.·26-s − 157.·28-s − 59.4·29-s + 40.2·31-s − 197.·32-s + 77.3·34-s − 505.·35-s + 184.·37-s − 487.·38-s − 173.·40-s − 368.·41-s − 407.·43-s − 17.7·46-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.620·4-s + 1.42·5-s − 1.71·7-s − 0.482·8-s + 1.81·10-s + 0.879·13-s − 2.18·14-s − 1.23·16-s + 0.306·17-s − 1.63·19-s + 0.884·20-s − 0.0446·23-s + 1.03·25-s + 1.11·26-s − 1.06·28-s − 0.380·29-s + 0.233·31-s − 1.09·32-s + 0.390·34-s − 2.43·35-s + 0.818·37-s − 2.07·38-s − 0.687·40-s − 1.40·41-s − 1.44·43-s − 0.0568·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.60T + 8T^{2} \) |
| 5 | \( 1 - 15.9T + 125T^{2} \) |
| 7 | \( 1 + 31.7T + 343T^{2} \) |
| 13 | \( 1 - 41.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 4.92T + 1.21e4T^{2} \) |
| 29 | \( 1 + 59.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 40.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 184.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 368.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 524.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 427.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 434.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 141.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 5.94T + 3.00e5T^{2} \) |
| 71 | \( 1 - 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 420.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 366.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 773.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 14.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−9.282639753341904394547409961923, −8.426715560137539716120085171272, −6.65435001862456106891656852281, −6.35646284824790787271019537172, −5.79613606434435075327341393875, −4.77553401722419823263347579535, −3.63586587154152252629071295738, −2.96102722869322841035465822634, −1.85175852664316128757711241550, 0,
1.85175852664316128757711241550, 2.96102722869322841035465822634, 3.63586587154152252629071295738, 4.77553401722419823263347579535, 5.79613606434435075327341393875, 6.35646284824790787271019537172, 6.65435001862456106891656852281, 8.426715560137539716120085171272, 9.282639753341904394547409961923
