L(s) = 1 | − 1.83·2-s + 3-s + 1.38·4-s + 0.446·5-s − 1.83·6-s + 3.21·7-s + 1.13·8-s + 9-s − 0.821·10-s + 0.755·11-s + 1.38·12-s − 6.00·13-s − 5.90·14-s + 0.446·15-s − 4.85·16-s − 17-s − 1.83·18-s − 3.47·19-s + 0.617·20-s + 3.21·21-s − 1.38·22-s − 1.33·23-s + 1.13·24-s − 4.80·25-s + 11.0·26-s + 27-s + 4.44·28-s + ⋯ |
L(s) = 1 | − 1.30·2-s + 0.577·3-s + 0.692·4-s + 0.199·5-s − 0.751·6-s + 1.21·7-s + 0.400·8-s + 0.333·9-s − 0.259·10-s + 0.227·11-s + 0.399·12-s − 1.66·13-s − 1.57·14-s + 0.115·15-s − 1.21·16-s − 0.242·17-s − 0.433·18-s − 0.797·19-s + 0.138·20-s + 0.700·21-s − 0.296·22-s − 0.278·23-s + 0.231·24-s − 0.960·25-s + 2.16·26-s + 0.192·27-s + 0.839·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 0.446T + 5T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 - 0.755T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 - 0.937T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 4.83T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 4.65T + 47T^{2} \) |
| 53 | \( 1 + 0.522T + 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 + 3.89T + 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 1.35T + 89T^{2} \) |
| 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
−7.56931560933043693148899571319, −7.41505802358419792773969132713, −6.30789579479806921501052756579, −5.39041617516468384171748731186, −4.45644760742346259025209058768, −4.14773461136695656995704207785, −2.53019979442147662696001215694, −2.13317599650542837055989698963, −1.27015701863682697104644227239, 0,
1.27015701863682697104644227239, 2.13317599650542837055989698963, 2.53019979442147662696001215694, 4.14773461136695656995704207785, 4.45644760742346259025209058768, 5.39041617516468384171748731186, 6.30789579479806921501052756579, 7.41505802358419792773969132713, 7.56931560933043693148899571319