| L(s) = 1 | + 2-s + 4-s − 0.686·5-s + 2.87·7-s + 8-s − 0.686·10-s − 5.51·11-s + 0.329·13-s + 2.87·14-s + 16-s − 4.44·17-s − 1.53·19-s − 0.686·20-s − 5.51·22-s + 3.17·23-s − 4.52·25-s + 0.329·26-s + 2.87·28-s − 7.03·29-s − 7.43·31-s + 32-s − 4.44·34-s − 1.97·35-s − 7.58·37-s − 1.53·38-s − 0.686·40-s + 2.51·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.306·5-s + 1.08·7-s + 0.353·8-s − 0.217·10-s − 1.66·11-s + 0.0914·13-s + 0.768·14-s + 0.250·16-s − 1.07·17-s − 0.352·19-s − 0.153·20-s − 1.17·22-s + 0.661·23-s − 0.905·25-s + 0.0646·26-s + 0.543·28-s − 1.30·29-s − 1.33·31-s + 0.176·32-s − 0.761·34-s − 0.333·35-s − 1.24·37-s − 0.249·38-s − 0.108·40-s + 0.393·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 0.686T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 0.329T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 + 7.03T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 - 2.51T + 41T^{2} \) |
| 43 | \( 1 - 6.97T + 43T^{2} \) |
| 47 | \( 1 + 6.00T + 47T^{2} \) |
| 53 | \( 1 - 0.840T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 5.22T + 83T^{2} \) |
| 89 | \( 1 + 8.29T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
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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
−7.82046288614393937529062494565, −7.52147501100821437939762238140, −6.58918930230629009373108453835, −5.51076514739119211144986029124, −5.16203934471662344695036348283, −4.34232724473818014368158973394, −3.54729462559134042702247929972, −2.43329179631741334718209031751, −1.78239988901328000640044416830, 0,
1.78239988901328000640044416830, 2.43329179631741334718209031751, 3.54729462559134042702247929972, 4.34232724473818014368158973394, 5.16203934471662344695036348283, 5.51076514739119211144986029124, 6.58918930230629009373108453835, 7.52147501100821437939762238140, 7.82046288614393937529062494565
