| L(s) = 1 | + 2-s − 2.92·3-s + 4-s + 3.92·5-s − 2.92·6-s − 7-s + 8-s + 5.55·9-s + 3.92·10-s − 4.55·11-s − 2.92·12-s + 3.55·13-s − 14-s − 11.4·15-s + 16-s − 3.63·17-s + 5.55·18-s − 7.48·19-s + 3.92·20-s + 2.92·21-s − 4.55·22-s − 2.92·24-s + 10.4·25-s + 3.55·26-s − 7.48·27-s − 28-s + 2.48·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.68·3-s + 0.5·4-s + 1.75·5-s − 1.19·6-s − 0.377·7-s + 0.353·8-s + 1.85·9-s + 1.24·10-s − 1.37·11-s − 0.844·12-s + 0.986·13-s − 0.267·14-s − 2.96·15-s + 0.250·16-s − 0.881·17-s + 1.31·18-s − 1.71·19-s + 0.877·20-s + 0.638·21-s − 0.971·22-s − 0.597·24-s + 2.08·25-s + 0.697·26-s − 1.44·27-s − 0.188·28-s + 0.461·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 - 3.92T + 5T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 3.63T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 0.292T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 8.77T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 - 0.483T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 9.29T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 9.43T + 97T^{2} \) |
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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
−6.99552205596874142890655255435, −6.41670565207748725546140343958, −6.08441627455990855397765006186, −5.65158272146840447568312785210, −4.77616767727702723406025445157, −4.48508615869088487041972114189, −3.03272789100574831606456072303, −2.19534051915074255172082501403, −1.37065561833906343849554782816, 0,
1.37065561833906343849554782816, 2.19534051915074255172082501403, 3.03272789100574831606456072303, 4.48508615869088487041972114189, 4.77616767727702723406025445157, 5.65158272146840447568312785210, 6.08441627455990855397765006186, 6.41670565207748725546140343958, 6.99552205596874142890655255435
