L(s) = 1 | − 2-s + 3.19·3-s + 4-s + 0.504·5-s − 3.19·6-s + 1.26·7-s − 8-s + 7.21·9-s − 0.504·10-s + 11-s + 3.19·12-s + 2.08·13-s − 1.26·14-s + 1.61·15-s + 16-s − 1.07·17-s − 7.21·18-s − 4.85·19-s + 0.504·20-s + 4.05·21-s − 22-s + 5.58·23-s − 3.19·24-s − 4.74·25-s − 2.08·26-s + 13.4·27-s + 1.26·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.84·3-s + 0.5·4-s + 0.225·5-s − 1.30·6-s + 0.479·7-s − 0.353·8-s + 2.40·9-s − 0.159·10-s + 0.301·11-s + 0.922·12-s + 0.579·13-s − 0.338·14-s + 0.416·15-s + 0.250·16-s − 0.260·17-s − 1.69·18-s − 1.11·19-s + 0.112·20-s + 0.884·21-s − 0.213·22-s + 1.16·23-s − 0.652·24-s − 0.949·25-s − 0.409·26-s + 2.59·27-s + 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.445459123\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.445459123\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - 3.19T + 3T^{2} \) |
| 5 | \( 1 - 0.504T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 - 0.301T + 53T^{2} \) |
| 59 | \( 1 + 0.968T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 67 | \( 1 - 9.93T + 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−9.669245314167262710029349644361, −9.030780016858690946073248862101, −8.608292462464324899488336878989, −7.69442492586968822432095903430, −7.14305321824142616116481279333, −5.94022332159658580577043393231, −4.37064344091257986565010125076, −3.49902244867315532268109819356, −2.34637387454212715395801863380, −1.55194407195367212266255749446,
1.55194407195367212266255749446, 2.34637387454212715395801863380, 3.49902244867315532268109819356, 4.37064344091257986565010125076, 5.94022332159658580577043393231, 7.14305321824142616116481279333, 7.69442492586968822432095903430, 8.608292462464324899488336878989, 9.030780016858690946073248862101, 9.669245314167262710029349644361