| L(s) = 1 | + 2.10·3-s + 5-s − 2.43·7-s + 1.43·9-s + 1.36·11-s − 5.47·13-s + 2.10·15-s − 3.28·17-s − 19-s − 5.14·21-s − 3.41·23-s + 25-s − 3.28·27-s − 0.926·29-s − 11.0·31-s + 2.87·33-s − 2.43·35-s − 6.90·37-s − 11.5·39-s + 4.21·41-s − 0.486·43-s + 1.43·45-s + 10.4·47-s − 1.04·49-s − 6.92·51-s + 13.2·53-s + 1.36·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 0.447·5-s − 0.922·7-s + 0.479·9-s + 0.412·11-s − 1.51·13-s + 0.544·15-s − 0.797·17-s − 0.229·19-s − 1.12·21-s − 0.711·23-s + 0.200·25-s − 0.632·27-s − 0.172·29-s − 1.99·31-s + 0.501·33-s − 0.412·35-s − 1.13·37-s − 1.84·39-s + 0.658·41-s − 0.0742·43-s + 0.214·45-s + 1.52·47-s − 0.149·49-s − 0.969·51-s + 1.81·53-s + 0.184·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 0.926T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 - 4.21T + 41T^{2} \) |
| 43 | \( 1 + 0.486T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.48T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 6.97T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 - 8.06T + 89T^{2} \) |
| 97 | \( 1 + 0.546T + 97T^{2} \) |
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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
−8.532664972048128315517283624135, −7.49793292272877085408005402381, −7.07021213015378097213692380747, −6.12794881462548803274027333251, −5.29056467951939825711015469880, −4.17172682693475938866123237867, −3.45010282894206533199980900888, −2.50563134851086477256336749490, −1.96057417664517149541423536885, 0,
1.96057417664517149541423536885, 2.50563134851086477256336749490, 3.45010282894206533199980900888, 4.17172682693475938866123237867, 5.29056467951939825711015469880, 6.12794881462548803274027333251, 7.07021213015378097213692380747, 7.49793292272877085408005402381, 8.532664972048128315517283624135
