| L(s) = 1 | − 3-s + 5-s + 9-s − 4.24·11-s − 3.24·13-s − 15-s − 4.24·17-s + 7·19-s − 4.24·23-s + 25-s − 27-s − 1.75·29-s + 9.48·31-s + 4.24·33-s + 3.24·37-s + 3.24·39-s + 4.24·41-s + 3.24·43-s + 45-s + 6·47-s + 4.24·51-s + 8.48·53-s − 4.24·55-s − 7·57-s + 10.2·59-s − 4.48·61-s − 3.24·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.333·9-s − 1.27·11-s − 0.899·13-s − 0.258·15-s − 1.02·17-s + 1.60·19-s − 0.884·23-s + 0.200·25-s − 0.192·27-s − 0.326·29-s + 1.70·31-s + 0.738·33-s + 0.533·37-s + 0.519·39-s + 0.662·41-s + 0.494·43-s + 0.149·45-s + 0.875·47-s + 0.594·51-s + 1.16·53-s − 0.572·55-s − 0.927·57-s + 1.33·59-s − 0.574·61-s − 0.402·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277660555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277660555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 0.485T + 97T^{2} \) |
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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
−8.832088134299587479808735352102, −7.75280475474106150398981478020, −7.37867433518451190658739974870, −6.35750558229820039477154041208, −5.64260900650721272230146777698, −4.99154067030928549846332516654, −4.24689803731577018075323963868, −2.87621076877478846564015366075, −2.18457316782557858845598713247, −0.69342725219036995367951150539,
0.69342725219036995367951150539, 2.18457316782557858845598713247, 2.87621076877478846564015366075, 4.24689803731577018075323963868, 4.99154067030928549846332516654, 5.64260900650721272230146777698, 6.35750558229820039477154041208, 7.37867433518451190658739974870, 7.75280475474106150398981478020, 8.832088134299587479808735352102
