| L(s) = 1 | − 3-s − 2·4-s + 5-s − 7-s + 9-s + 6·11-s + 2·12-s − 13-s − 15-s + 4·16-s + 2·19-s − 2·20-s + 21-s − 3·23-s + 25-s − 27-s + 2·28-s − 5·31-s − 6·33-s − 35-s − 2·36-s − 5·37-s + 39-s − 9·41-s + 8·43-s − 12·44-s + 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s + 0.458·19-s − 0.447·20-s + 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.898·31-s − 1.04·33-s − 0.169·35-s − 1/3·36-s − 0.821·37-s + 0.160·39-s − 1.40·41-s + 1.21·43-s − 1.80·44-s + 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.317238710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317238710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.02324983666994, −14.37729851816171, −14.12996877771057, −13.60267617305004, −12.96071585858208, −12.47174480593677, −11.99014829761875, −11.52745513991189, −10.79116692558634, −10.12358874902941, −9.704076825957254, −9.217280630409446, −8.835866009675682, −8.087950325434656, −7.339988824082485, −6.684397046653496, −6.236243327091405, −5.588440932616596, −5.024499625622982, −4.396480383746158, −3.686230658250724, −3.330497507804493, −2.011935853026733, −1.347902025215126, −0.4978907039231533,
0.4978907039231533, 1.347902025215126, 2.011935853026733, 3.330497507804493, 3.686230658250724, 4.396480383746158, 5.024499625622982, 5.588440932616596, 6.236243327091405, 6.684397046653496, 7.339988824082485, 8.087950325434656, 8.835866009675682, 9.217280630409446, 9.704076825957254, 10.12358874902941, 10.79116692558634, 11.52745513991189, 11.99014829761875, 12.47174480593677, 12.96071585858208, 13.60267617305004, 14.12996877771057, 14.37729851816171, 15.02324983666994
