| L(s) = 1 | + 3·3-s − 5·5-s − 20·7-s + 9·9-s + 24·11-s + 74·13-s − 15·15-s + 54·17-s + 124·19-s − 60·21-s + 120·23-s + 25·25-s + 27·27-s − 78·29-s − 200·31-s + 72·33-s + 100·35-s − 70·37-s + 222·39-s + 330·41-s − 92·43-s − 45·45-s + 24·47-s + 57·49-s + 162·51-s + 450·53-s − 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.07·7-s + 1/3·9-s + 0.657·11-s + 1.57·13-s − 0.258·15-s + 0.770·17-s + 1.49·19-s − 0.623·21-s + 1.08·23-s + 1/5·25-s + 0.192·27-s − 0.499·29-s − 1.15·31-s + 0.379·33-s + 0.482·35-s − 0.311·37-s + 0.911·39-s + 1.25·41-s − 0.326·43-s − 0.149·45-s + 0.0744·47-s + 0.166·49-s + 0.444·51-s + 1.16·53-s − 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.052434277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052434277\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 - 196 T + p^{3} T^{2} \) |
| 71 | \( 1 - 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−11.70514687402083509794356219818, −10.73263158013076342300041175229, −9.500705683415271053710452125197, −8.937592327110968691338678283199, −7.70687932906837603038162066525, −6.75820693404535152512177285178, −5.58622918905472991418170808513, −3.80832995564793852354231988723, −3.18913826885681635662080465351, −1.11021679759684095895145142700,
1.11021679759684095895145142700, 3.18913826885681635662080465351, 3.80832995564793852354231988723, 5.58622918905472991418170808513, 6.75820693404535152512177285178, 7.70687932906837603038162066525, 8.937592327110968691338678283199, 9.500705683415271053710452125197, 10.73263158013076342300041175229, 11.70514687402083509794356219818
