| L(s) = 1 | + 1.60·2-s + 0.568·4-s + 1.76·5-s − 7-s − 2.29·8-s + 2.82·10-s + 0.237·13-s − 1.60·14-s − 4.81·16-s − 6.14·17-s + 1.53·19-s + 1.00·20-s + 3.89·23-s − 1.88·25-s + 0.381·26-s − 0.568·28-s + 4.45·29-s + 8.52·31-s − 3.12·32-s − 9.84·34-s − 1.76·35-s − 3.64·37-s + 2.46·38-s − 4.04·40-s + 1.90·41-s − 11.7·43-s + 6.24·46-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.284·4-s + 0.789·5-s − 0.377·7-s − 0.811·8-s + 0.894·10-s + 0.0659·13-s − 0.428·14-s − 1.20·16-s − 1.48·17-s + 0.352·19-s + 0.224·20-s + 0.812·23-s − 0.377·25-s + 0.0747·26-s − 0.107·28-s + 0.827·29-s + 1.53·31-s − 0.552·32-s − 1.68·34-s − 0.298·35-s − 0.599·37-s + 0.399·38-s − 0.640·40-s + 0.297·41-s − 1.78·43-s + 0.920·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 13 | \( 1 - 0.237T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 - 8.52T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 - 1.90T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 6.78T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−7.18890934170274945630251420336, −6.47451704031338573334438516558, −6.19380045917005253213414094754, −5.27482809633201625562793920831, −4.74895546887225850765117528286, −4.08309449925607705479628995574, −3.08905671092872305745445390301, −2.60944891874790410328853462814, −1.51285815626070692238415219090, 0,
1.51285815626070692238415219090, 2.60944891874790410328853462814, 3.08905671092872305745445390301, 4.08309449925607705479628995574, 4.74895546887225850765117528286, 5.27482809633201625562793920831, 6.19380045917005253213414094754, 6.47451704031338573334438516558, 7.18890934170274945630251420336
