L(s) = 1 | + 2-s − 1.87·3-s + 4-s + 0.417·5-s − 1.87·6-s − 3.88·7-s + 8-s + 0.514·9-s + 0.417·10-s + 1.68·11-s − 1.87·12-s + 0.977·13-s − 3.88·14-s − 0.782·15-s + 16-s + 4.13·17-s + 0.514·18-s + 0.960·19-s + 0.417·20-s + 7.28·21-s + 1.68·22-s − 23-s − 1.87·24-s − 4.82·25-s + 0.977·26-s + 4.65·27-s − 3.88·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.08·3-s + 0.5·4-s + 0.186·5-s − 0.765·6-s − 1.46·7-s + 0.353·8-s + 0.171·9-s + 0.131·10-s + 0.508·11-s − 0.541·12-s + 0.271·13-s − 1.03·14-s − 0.201·15-s + 0.250·16-s + 1.00·17-s + 0.121·18-s + 0.220·19-s + 0.0932·20-s + 1.58·21-s + 0.359·22-s − 0.208·23-s − 0.382·24-s − 0.965·25-s + 0.191·26-s + 0.896·27-s − 0.734·28-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 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 0.417T + 5T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 0.977T + 13T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 - 0.960T + 19T^{2} \) |
| 29 | \( 1 - 2.28T + 29T^{2} \) |
| 31 | \( 1 + 5.85T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 - 4.46T + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 1.30T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 - 3.02T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−7.39914619508906832602932437984, −6.64262362617490786886144608791, −6.25608448989188431111522789401, −5.58449512653730312607971106786, −5.13285553245779407727841735759, −3.89786054815545765111285828997, −3.48759485156334037676931340640, −2.50606952528396507056127425044, −1.21625359395333293795086999634, 0,
1.21625359395333293795086999634, 2.50606952528396507056127425044, 3.48759485156334037676931340640, 3.89786054815545765111285828997, 5.13285553245779407727841735759, 5.58449512653730312607971106786, 6.25608448989188431111522789401, 6.64262362617490786886144608791, 7.39914619508906832602932437984