| L(s) = 1 | + 2-s + 4-s + 3.92·5-s + 3.81·7-s + 8-s + 3.92·10-s − 0.0352·11-s + 1.31·13-s + 3.81·14-s + 16-s + 2.23·17-s + 1.55·19-s + 3.92·20-s − 0.0352·22-s − 8.61·23-s + 10.3·25-s + 1.31·26-s + 3.81·28-s − 3.50·29-s + 7.40·31-s + 32-s + 2.23·34-s + 14.9·35-s − 7.32·37-s + 1.55·38-s + 3.92·40-s − 4.54·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.75·5-s + 1.44·7-s + 0.353·8-s + 1.24·10-s − 0.0106·11-s + 0.365·13-s + 1.01·14-s + 0.250·16-s + 0.541·17-s + 0.357·19-s + 0.877·20-s − 0.00751·22-s − 1.79·23-s + 2.07·25-s + 0.258·26-s + 0.720·28-s − 0.650·29-s + 1.33·31-s + 0.176·32-s + 0.382·34-s + 2.52·35-s − 1.20·37-s + 0.253·38-s + 0.620·40-s − 0.709·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.265084938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.265084938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 3.92T + 5T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 + 0.0352T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 + 8.61T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 7.40T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 + 8.28T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 + 6.33T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 - 6.54T + 73T^{2} \) |
| 79 | \( 1 - 9.99T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−8.262415566652809503592378847469, −7.82255016186317405998141179545, −6.66657307094869102413393790837, −6.10401368134791903815458930113, −5.35169554674340126387741028422, −4.96384903043501307647237734470, −3.96306071066342724923449057959, −2.84145075601348662891347374976, −1.85561723315999398616550888436, −1.46142672928273964458415285548,
1.46142672928273964458415285548, 1.85561723315999398616550888436, 2.84145075601348662891347374976, 3.96306071066342724923449057959, 4.96384903043501307647237734470, 5.35169554674340126387741028422, 6.10401368134791903815458930113, 6.66657307094869102413393790837, 7.82255016186317405998141179545, 8.262415566652809503592378847469
