| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 1.80·5-s − 6·6-s − 16.5·7-s − 8·8-s + 9·9-s − 3.61·10-s + 42.8·11-s + 12·12-s − 69.9·13-s + 33.0·14-s + 5.42·15-s + 16·16-s + 67.0·17-s − 18·18-s + 7.23·20-s − 49.6·21-s − 85.7·22-s + 97.4·23-s − 24·24-s − 121.·25-s + 139.·26-s + 27·27-s − 66.1·28-s − 45.6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.161·5-s − 0.408·6-s − 0.892·7-s − 0.353·8-s + 0.333·9-s − 0.114·10-s + 1.17·11-s + 0.288·12-s − 1.49·13-s + 0.631·14-s + 0.0934·15-s + 0.250·16-s + 0.955·17-s − 0.235·18-s + 0.0809·20-s − 0.515·21-s − 0.831·22-s + 0.883·23-s − 0.204·24-s − 0.973·25-s + 1.05·26-s + 0.192·27-s − 0.446·28-s − 0.292·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 1.80T + 125T^{2} \) |
| 7 | \( 1 + 16.5T + 343T^{2} \) |
| 11 | \( 1 - 42.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 69.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 97.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 45.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 91.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 641.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 238.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 462.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 634.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 684.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 974.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 495.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 411.T + 9.12e5T^{2} \) |
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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
−8.429910263497854732824797446113, −7.50634779939703445696685316982, −7.02596003422400950496841995120, −6.19205804231621215247025348735, −5.23801473940654947095951137905, −4.00113979919526935725789079514, −3.18871954419049265616045532512, −2.29695145086973013850605585526, −1.23326834659100325912651918144, 0,
1.23326834659100325912651918144, 2.29695145086973013850605585526, 3.18871954419049265616045532512, 4.00113979919526935725789079514, 5.23801473940654947095951137905, 6.19205804231621215247025348735, 7.02596003422400950496841995120, 7.50634779939703445696685316982, 8.429910263497854732824797446113
