| L(s) = 1 | + 2-s + 0.334·3-s + 4-s − 2.22·5-s + 0.334·6-s − 7-s + 8-s − 2.88·9-s − 2.22·10-s + 0.585·11-s + 0.334·12-s − 5.97·13-s − 14-s − 0.744·15-s + 16-s + 4.22·17-s − 2.88·18-s + 2.66·19-s − 2.22·20-s − 0.334·21-s + 0.585·22-s + 0.334·24-s − 0.0594·25-s − 5.97·26-s − 1.97·27-s − 28-s − 4.43·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.193·3-s + 0.5·4-s − 0.994·5-s + 0.136·6-s − 0.377·7-s + 0.353·8-s − 0.962·9-s − 0.702·10-s + 0.176·11-s + 0.0966·12-s − 1.65·13-s − 0.267·14-s − 0.192·15-s + 0.250·16-s + 1.02·17-s − 0.680·18-s + 0.612·19-s − 0.497·20-s − 0.0730·21-s + 0.124·22-s + 0.0683·24-s − 0.0118·25-s − 1.17·26-s − 0.379·27-s − 0.188·28-s − 0.824·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.832317780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832317780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.334T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 - 6.46T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 9.97T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.64024322735909485982121981937, −7.38685037948276988005840288503, −6.49450584547489860153426700860, −5.62192843145477844341658946871, −5.10984088735742219999490265083, −4.27416015875402938522839564619, −3.45987467291543737831104210185, −2.98054158376695139819435748026, −2.10611567259679401947539429418, −0.57454605904393982169827302657,
0.57454605904393982169827302657, 2.10611567259679401947539429418, 2.98054158376695139819435748026, 3.45987467291543737831104210185, 4.27416015875402938522839564619, 5.10984088735742219999490265083, 5.62192843145477844341658946871, 6.49450584547489860153426700860, 7.38685037948276988005840288503, 7.64024322735909485982121981937
