| L(s) = 1 | − 2-s − 3-s + 4-s − 3.83·5-s + 6-s − 4.91·7-s − 8-s + 9-s + 3.83·10-s − 1.87·11-s − 12-s − 3.25·13-s + 4.91·14-s + 3.83·15-s + 16-s + 5.70·17-s − 18-s + 0.533·19-s − 3.83·20-s + 4.91·21-s + 1.87·22-s + 23-s + 24-s + 9.66·25-s + 3.25·26-s − 27-s − 4.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.71·5-s + 0.408·6-s − 1.85·7-s − 0.353·8-s + 0.333·9-s + 1.21·10-s − 0.565·11-s − 0.288·12-s − 0.902·13-s + 1.31·14-s + 0.988·15-s + 0.250·16-s + 1.38·17-s − 0.235·18-s + 0.122·19-s − 0.856·20-s + 1.07·21-s + 0.399·22-s + 0.208·23-s + 0.204·24-s + 1.93·25-s + 0.638·26-s − 0.192·27-s − 0.928·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.83T + 5T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 0.533T + 19T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 0.389T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 - 5.94T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 3.97T + 97T^{2} \) |
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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
−7.79357917869079747112791660928, −7.46690798627149827489981611666, −6.88880940860389292765666508539, −5.99497680174800263226631439144, −5.19973178676256703420242350738, −4.05922036847546125271315778246, −3.37038555009085593767654404312, −2.66852703748557789518742391329, −0.77195571757687549539519268810, 0,
0.77195571757687549539519268810, 2.66852703748557789518742391329, 3.37038555009085593767654404312, 4.05922036847546125271315778246, 5.19973178676256703420242350738, 5.99497680174800263226631439144, 6.88880940860389292765666508539, 7.46690798627149827489981611666, 7.79357917869079747112791660928
