| L(s) = 1 | + 2-s + 2.96·3-s + 4-s + 1.05·5-s + 2.96·6-s − 1.88·7-s + 8-s + 5.79·9-s + 1.05·10-s − 11-s + 2.96·12-s + 4.10·13-s − 1.88·14-s + 3.12·15-s + 16-s − 1.43·17-s + 5.79·18-s − 2.65·19-s + 1.05·20-s − 5.59·21-s − 22-s + 1.83·23-s + 2.96·24-s − 3.88·25-s + 4.10·26-s + 8.27·27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s + 0.471·5-s + 1.21·6-s − 0.713·7-s + 0.353·8-s + 1.93·9-s + 0.333·10-s − 0.301·11-s + 0.855·12-s + 1.13·13-s − 0.504·14-s + 0.807·15-s + 0.250·16-s − 0.348·17-s + 1.36·18-s − 0.608·19-s + 0.235·20-s − 1.22·21-s − 0.213·22-s + 0.382·23-s + 0.605·24-s − 0.777·25-s + 0.805·26-s + 1.59·27-s − 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.208907001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.208907001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 + 0.115T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 - 8.72T + 43T^{2} \) |
| 47 | \( 1 + 0.0926T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 67 | \( 1 - 0.148T + 67T^{2} \) |
| 71 | \( 1 + 9.69T + 71T^{2} \) |
| 73 | \( 1 - 3.05T + 73T^{2} \) |
| 79 | \( 1 - 9.39T + 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 - 7.68T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.336993756368819652512658692444, −7.78279373656394725652215429191, −6.82453096765022631584269668374, −6.30384129108610844496278270474, −5.41517531000143348837176005829, −4.19073318838611812586764935512, −3.83645204554802214817676324529, −2.79359196101470742526385469597, −2.44922386053687986388938387280, −1.29555807287720803289564191207,
1.29555807287720803289564191207, 2.44922386053687986388938387280, 2.79359196101470742526385469597, 3.83645204554802214817676324529, 4.19073318838611812586764935512, 5.41517531000143348837176005829, 6.30384129108610844496278270474, 6.82453096765022631584269668374, 7.78279373656394725652215429191, 8.336993756368819652512658692444
