| L(s) = 1 | + 5.49·2-s + 3·3-s + 22.2·4-s + 5·5-s + 16.4·6-s + 22.4·7-s + 78.0·8-s + 9·9-s + 27.4·10-s + 25.1·11-s + 66.6·12-s − 37.6·13-s + 123.·14-s + 15·15-s + 251.·16-s − 106.·17-s + 49.4·18-s + 112.·19-s + 111.·20-s + 67.2·21-s + 138.·22-s − 20.3·23-s + 234.·24-s + 25·25-s − 206.·26-s + 27·27-s + 497.·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.77·4-s + 0.447·5-s + 1.12·6-s + 1.21·7-s + 3.44·8-s + 0.333·9-s + 0.868·10-s + 0.689·11-s + 1.60·12-s − 0.803·13-s + 2.35·14-s + 0.258·15-s + 3.92·16-s − 1.51·17-s + 0.647·18-s + 1.36·19-s + 1.24·20-s + 0.698·21-s + 1.33·22-s − 0.184·23-s + 1.99·24-s + 0.200·25-s − 1.56·26-s + 0.192·27-s + 3.35·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.63711250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.63711250\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 163 | \( 1 + 163T \) |
| good | 2 | \( 1 - 5.49T + 8T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 11 | \( 1 - 25.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 20.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 231.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 209.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 313.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 484.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 880.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 508.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 759.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 182.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 695.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 594.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 684.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 546.T + 9.12e5T^{2} \) |
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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
−8.379100710152147754416291594326, −7.43017394767259080888792357009, −6.98231411172425502820573503077, −6.04701256718448769631607650623, −5.18073300562855831978166884473, −4.65151178790300962164908839145, −3.91794707051004987707832341106, −2.93758838417411514233423191214, −2.08216048601472670528721757341, −1.47391753193003520333659822037,
1.47391753193003520333659822037, 2.08216048601472670528721757341, 2.93758838417411514233423191214, 3.91794707051004987707832341106, 4.65151178790300962164908839145, 5.18073300562855831978166884473, 6.04701256718448769631607650623, 6.98231411172425502820573503077, 7.43017394767259080888792357009, 8.379100710152147754416291594326
