L(s) = 1 | − 3-s + 0.554·5-s + 3.04·7-s + 9-s − 1.80·11-s − 0.554·15-s + 1.24·17-s − 2.35·19-s − 3.04·21-s − 0.554·23-s − 4.69·25-s − 27-s − 6.18·29-s − 8.67·31-s + 1.80·33-s + 1.69·35-s − 0.960·37-s + 2.47·41-s + 0.384·43-s + 0.554·45-s − 9.96·47-s + 2.29·49-s − 1.24·51-s + 6.02·53-s − 55-s + 2.35·57-s + 7.30·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.248·5-s + 1.15·7-s + 0.333·9-s − 0.543·11-s − 0.143·15-s + 0.302·17-s − 0.540·19-s − 0.665·21-s − 0.115·23-s − 0.938·25-s − 0.192·27-s − 1.14·29-s − 1.55·31-s + 0.313·33-s + 0.286·35-s − 0.157·37-s + 0.386·41-s + 0.0585·43-s + 0.0827·45-s − 1.45·47-s + 0.327·49-s − 0.174·51-s + 0.827·53-s − 0.134·55-s + 0.312·57-s + 0.951·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.554T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 0.554T + 23T^{2} \) |
| 29 | \( 1 + 6.18T + 29T^{2} \) |
| 31 | \( 1 + 8.67T + 31T^{2} \) |
| 37 | \( 1 + 0.960T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 0.384T + 43T^{2} \) |
| 47 | \( 1 + 9.96T + 47T^{2} \) |
| 53 | \( 1 - 6.02T + 53T^{2} \) |
| 59 | \( 1 - 7.30T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 4.58T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.81122922121505991761865822224, −7.57973354855928997071468715333, −6.51717320576126560707354994941, −5.67370578175794414336025082166, −5.21924205262885671489146861984, −4.39563685054332673008297136558, −3.53871096865880968370287696103, −2.19276631741628805320952165008, −1.52143726837963982112158615223, 0,
1.52143726837963982112158615223, 2.19276631741628805320952165008, 3.53871096865880968370287696103, 4.39563685054332673008297136558, 5.21924205262885671489146861984, 5.67370578175794414336025082166, 6.51717320576126560707354994941, 7.57973354855928997071468715333, 7.81122922121505991761865822224