| L(s) = 1 | − 2.93·2-s + 3·3-s + 0.630·4-s − 8.81·6-s + 18.9·7-s + 21.6·8-s + 9·9-s − 6.06·11-s + 1.89·12-s + 78.2·13-s − 55.5·14-s − 68.6·16-s + 38.4·17-s − 26.4·18-s − 92.9·19-s + 56.7·21-s + 17.8·22-s − 81.0·23-s + 64.9·24-s − 229.·26-s + 27·27-s + 11.9·28-s − 11.1·29-s − 223.·31-s + 28.4·32-s − 18.2·33-s − 112.·34-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.577·3-s + 0.0788·4-s − 0.599·6-s + 1.02·7-s + 0.956·8-s + 0.333·9-s − 0.166·11-s + 0.0455·12-s + 1.66·13-s − 1.06·14-s − 1.07·16-s + 0.547·17-s − 0.346·18-s − 1.12·19-s + 0.589·21-s + 0.172·22-s − 0.734·23-s + 0.552·24-s − 1.73·26-s + 0.192·27-s + 0.0805·28-s − 0.0714·29-s − 1.29·31-s + 0.157·32-s − 0.0960·33-s − 0.569·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.93T + 8T^{2} \) |
| 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 + 6.06T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 81.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 107.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 350.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 291.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 684.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 223.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 697.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 909.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 291.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 709.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 3.77T + 7.04e5T^{2} \) |
| 97 | \( 1 - 18.4T + 9.12e5T^{2} \) |
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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
−8.552524288402296319920664915730, −8.036344299624704961188254129486, −7.28512774116924520580698225083, −6.24817942923156069020841561334, −5.17173263570908966620922184695, −4.24233438986693811896093897335, −3.43790659197265624820137947886, −1.86023140571303293097269545125, −1.42202897673161948304178206609, 0,
1.42202897673161948304178206609, 1.86023140571303293097269545125, 3.43790659197265624820137947886, 4.24233438986693811896093897335, 5.17173263570908966620922184695, 6.24817942923156069020841561334, 7.28512774116924520580698225083, 8.036344299624704961188254129486, 8.552524288402296319920664915730
