| L(s) = 1 | − 2-s − 1.90·3-s + 4-s − 1.40·5-s + 1.90·6-s − 1.81·7-s − 8-s + 0.624·9-s + 1.40·10-s + 1.73·11-s − 1.90·12-s − 2.64·13-s + 1.81·14-s + 2.67·15-s + 16-s + 1.78·17-s − 0.624·18-s − 2.37·19-s − 1.40·20-s + 3.45·21-s − 1.73·22-s − 23-s + 1.90·24-s − 3.03·25-s + 2.64·26-s + 4.52·27-s − 1.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.09·3-s + 0.5·4-s − 0.627·5-s + 0.777·6-s − 0.686·7-s − 0.353·8-s + 0.208·9-s + 0.443·10-s + 0.523·11-s − 0.549·12-s − 0.732·13-s + 0.485·14-s + 0.689·15-s + 0.250·16-s + 0.432·17-s − 0.147·18-s − 0.545·19-s − 0.313·20-s + 0.754·21-s − 0.370·22-s − 0.208·23-s + 0.388·24-s − 0.606·25-s + 0.518·26-s + 0.870·27-s − 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 29 | \( 1 - 0.0835T + 29T^{2} \) |
| 31 | \( 1 + 0.944T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 0.0652T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 8.91T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−7.79068828756058209737897483613, −6.91922033753621651530243668733, −6.38933122192595594776684358656, −5.79949143474706414870433813424, −4.93014156185797726675895872991, −4.07097831983168994400526039031, −3.20328813508364001012641708746, −2.18136598902064919066698814907, −0.851344867339571144973938938290, 0,
0.851344867339571144973938938290, 2.18136598902064919066698814907, 3.20328813508364001012641708746, 4.07097831983168994400526039031, 4.93014156185797726675895872991, 5.79949143474706414870433813424, 6.38933122192595594776684358656, 6.91922033753621651530243668733, 7.79068828756058209737897483613
