| L(s) = 1 | + 2·2-s − 4.84·3-s + 4·4-s − 6.31·5-s − 9.68·6-s + 8·8-s − 3.53·9-s − 12.6·10-s − 56.5·11-s − 19.3·12-s − 23.2·13-s + 30.5·15-s + 16·16-s − 44.2·17-s − 7.06·18-s + 82.8·19-s − 25.2·20-s − 113.·22-s − 23·23-s − 38.7·24-s − 85.1·25-s − 46.4·26-s + 147.·27-s − 283.·29-s + 61.1·30-s − 138.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.932·3-s + 0.5·4-s − 0.564·5-s − 0.659·6-s + 0.353·8-s − 0.130·9-s − 0.399·10-s − 1.54·11-s − 0.466·12-s − 0.495·13-s + 0.526·15-s + 0.250·16-s − 0.631·17-s − 0.0924·18-s + 1.00·19-s − 0.282·20-s − 1.09·22-s − 0.208·23-s − 0.329·24-s − 0.680·25-s − 0.350·26-s + 1.05·27-s − 1.81·29-s + 0.372·30-s − 0.802·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6057130224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6057130224\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 4.84T + 27T^{2} \) |
| 5 | \( 1 + 6.31T + 125T^{2} \) |
| 11 | \( 1 + 56.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 283.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 260.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 227.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 350.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 555.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 443.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 65.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 170.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 949.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 790.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−8.504302172633432659501184177065, −7.58656295382268659314161318671, −7.19423224181186384724502909800, −6.04652232049926100105295072536, −5.39507939632942998159881664529, −4.95293377928857727271285824763, −3.88616647316019798486797422833, −2.96896171077492401773945340289, −1.95298215396393311176257796198, −0.30846846758117163054101630710,
0.30846846758117163054101630710, 1.95298215396393311176257796198, 2.96896171077492401773945340289, 3.88616647316019798486797422833, 4.95293377928857727271285824763, 5.39507939632942998159881664529, 6.04652232049926100105295072536, 7.19423224181186384724502909800, 7.58656295382268659314161318671, 8.504302172633432659501184177065
