| L(s) = 1 | − 2.10·2-s + 2.42·4-s − 3.75·5-s + 7-s − 0.897·8-s + 7.89·10-s + 3.46·13-s − 2.10·14-s − 2.96·16-s + 1.52·17-s − 5.16·19-s − 9.10·20-s − 4.87·23-s + 9.06·25-s − 7.29·26-s + 2.42·28-s + 10.7·29-s + 8.12·31-s + 8.03·32-s − 3.21·34-s − 3.75·35-s + 9.25·37-s + 10.8·38-s + 3.36·40-s − 10.8·41-s − 0.137·43-s + 10.2·46-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.21·4-s − 1.67·5-s + 0.377·7-s − 0.317·8-s + 2.49·10-s + 0.961·13-s − 0.562·14-s − 0.741·16-s + 0.370·17-s − 1.18·19-s − 2.03·20-s − 1.01·23-s + 1.81·25-s − 1.43·26-s + 0.458·28-s + 1.99·29-s + 1.45·31-s + 1.42·32-s − 0.550·34-s − 0.633·35-s + 1.52·37-s + 1.76·38-s + 0.532·40-s − 1.70·41-s − 0.0209·43-s + 1.51·46-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
| \(L(1)\) |
\(\approx\) |
\(0.5643340971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5643340971\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.10T + 2T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 - 9.25T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 0.137T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 2.79T + 53T^{2} \) |
| 59 | \( 1 + 4.84T + 59T^{2} \) |
| 61 | \( 1 + 0.297T + 61T^{2} \) |
| 67 | \( 1 - 0.816T + 67T^{2} \) |
| 71 | \( 1 - 5.72T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 8.91T + 89T^{2} \) |
| 97 | \( 1 - 8.08T + 97T^{2} \) |
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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
−8.110274035467896902590668708259, −7.61950194970035651731346251900, −6.61177346237804640150826081636, −6.30292507463205724880256209367, −4.76902578164879719954889785418, −4.34455637577609572929518876032, −3.48858424453543946203715834244, −2.50710574313891375931095360820, −1.33346708239284496415964443138, −0.52634154631144165428305380113,
0.52634154631144165428305380113, 1.33346708239284496415964443138, 2.50710574313891375931095360820, 3.48858424453543946203715834244, 4.34455637577609572929518876032, 4.76902578164879719954889785418, 6.30292507463205724880256209367, 6.61177346237804640150826081636, 7.61950194970035651731346251900, 8.110274035467896902590668708259
