| L(s) = 1 | − 1.79·3-s + 5-s + 2.79·7-s + 0.208·9-s − 3.79·11-s − 1.20·13-s − 1.79·15-s − 3.79·17-s − 1.20·19-s − 5·21-s + 23-s + 25-s + 5.00·27-s + 1.58·29-s + 10.3·31-s + 6.79·33-s + 2.79·35-s + 4·37-s + 2.16·39-s − 2.20·41-s + 7.16·43-s + 0.208·45-s − 13.5·47-s + 0.791·49-s + 6.79·51-s − 6·53-s − 3.79·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 0.447·5-s + 1.05·7-s + 0.0695·9-s − 1.14·11-s − 0.335·13-s − 0.462·15-s − 0.919·17-s − 0.277·19-s − 1.09·21-s + 0.208·23-s + 0.200·25-s + 0.962·27-s + 0.293·29-s + 1.86·31-s + 1.18·33-s + 0.471·35-s + 0.657·37-s + 0.346·39-s − 0.344·41-s + 1.09·43-s + 0.0311·45-s − 1.98·47-s + 0.113·49-s + 0.950·51-s − 0.824·53-s − 0.511·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 7.16T + 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 3.16T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.61526191631514262568548275866, −6.59509008933691159367212668262, −6.21936048580436766920642169472, −5.27311148137433832417669718019, −4.92224953166670623042522099035, −4.34987330605809724263098116840, −2.89714219406380657439587058079, −2.26238286471411208383711708274, −1.15522517300381585841096040172, 0,
1.15522517300381585841096040172, 2.26238286471411208383711708274, 2.89714219406380657439587058079, 4.34987330605809724263098116840, 4.92224953166670623042522099035, 5.27311148137433832417669718019, 6.21936048580436766920642169472, 6.59509008933691159367212668262, 7.61526191631514262568548275866
