L(s) = 1 | − 3-s − 0.585·5-s + 1.41·7-s + 9-s − 0.828·11-s − 4.82·13-s + 0.585·15-s − 0.828·17-s + 2.82·19-s − 1.41·21-s + 6.82·23-s − 4.65·25-s − 27-s − 4.58·29-s − 7.07·31-s + 0.828·33-s − 0.828·35-s − 0.343·37-s + 4.82·39-s + 6.48·41-s − 1.17·43-s − 0.585·45-s − 4.48·47-s − 5·49-s + 0.828·51-s − 10.2·53-s + 0.485·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.261·5-s + 0.534·7-s + 0.333·9-s − 0.249·11-s − 1.33·13-s + 0.151·15-s − 0.200·17-s + 0.648·19-s − 0.308·21-s + 1.42·23-s − 0.931·25-s − 0.192·27-s − 0.851·29-s − 1.27·31-s + 0.144·33-s − 0.140·35-s − 0.0564·37-s + 0.773·39-s + 1.01·41-s − 0.178·43-s − 0.0873·45-s − 0.654·47-s − 0.714·49-s + 0.116·51-s − 1.40·53-s + 0.0654·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 4.58T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.343T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−9.295711083955550834583111827462, −8.036590783328742296371625071990, −7.45997117680876995555470128582, −6.75279722416123366354334408072, −5.51710148135906901587650738618, −5.04374812566538833340122756368, −4.09438262518389572972899910348, −2.87461434207493563218776657154, −1.62207441971997102714819009959, 0,
1.62207441971997102714819009959, 2.87461434207493563218776657154, 4.09438262518389572972899910348, 5.04374812566538833340122756368, 5.51710148135906901587650738618, 6.75279722416123366354334408072, 7.45997117680876995555470128582, 8.036590783328742296371625071990, 9.295711083955550834583111827462