| L(s) = 1 | − 0.434·2-s − 2.09·3-s − 1.81·4-s + 0.909·6-s − 3.42·7-s + 1.65·8-s + 1.37·9-s − 3.52·11-s + 3.78·12-s − 4.34·13-s + 1.49·14-s + 2.90·16-s − 0.147·17-s − 0.597·18-s − 8.29·19-s + 7.17·21-s + 1.53·22-s − 3.47·23-s − 3.46·24-s + 1.88·26-s + 3.39·27-s + 6.20·28-s − 10.4·29-s + 2.52·31-s − 4.57·32-s + 7.37·33-s + 0.0640·34-s + ⋯ |
| L(s) = 1 | − 0.307·2-s − 1.20·3-s − 0.905·4-s + 0.371·6-s − 1.29·7-s + 0.585·8-s + 0.458·9-s − 1.06·11-s + 1.09·12-s − 1.20·13-s + 0.398·14-s + 0.725·16-s − 0.0357·17-s − 0.140·18-s − 1.90·19-s + 1.56·21-s + 0.326·22-s − 0.724·23-s − 0.707·24-s + 0.370·26-s + 0.654·27-s + 1.17·28-s − 1.94·29-s + 0.453·31-s − 0.808·32-s + 1.28·33-s + 0.0109·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07957103207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07957103207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.434T + 2T^{2} \) |
| 3 | \( 1 + 2.09T + 3T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 0.147T + 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 + 0.513T + 41T^{2} \) |
| 47 | \( 1 + 6.75T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−10.07793581544158538710329040591, −9.246649336215538227560708559639, −8.277187013639932048441100543827, −7.34445001282737765453442391624, −6.36149780202960863645348419624, −5.61974706452576990733127491873, −4.82454558986269796187773858878, −3.86939411540228747406390720589, −2.41930392308318315368271912318, −0.22372593531266057849989057278,
0.22372593531266057849989057278, 2.41930392308318315368271912318, 3.86939411540228747406390720589, 4.82454558986269796187773858878, 5.61974706452576990733127491873, 6.36149780202960863645348419624, 7.34445001282737765453442391624, 8.277187013639932048441100543827, 9.246649336215538227560708559639, 10.07793581544158538710329040591
