| L(s) = 1 | − 2-s − 3·3-s − 7·4-s + 3·6-s + 24·7-s + 15·8-s + 9·9-s + 52·11-s + 21·12-s − 22·13-s − 24·14-s + 41·16-s + 14·17-s − 9·18-s − 20·19-s − 72·21-s − 52·22-s + 168·23-s − 45·24-s + 22·26-s − 27·27-s − 168·28-s + 230·29-s − 288·31-s − 161·32-s − 156·33-s − 14·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.204·6-s + 1.29·7-s + 0.662·8-s + 1/3·9-s + 1.42·11-s + 0.505·12-s − 0.469·13-s − 0.458·14-s + 0.640·16-s + 0.199·17-s − 0.117·18-s − 0.241·19-s − 0.748·21-s − 0.503·22-s + 1.52·23-s − 0.382·24-s + 0.165·26-s − 0.192·27-s − 1.13·28-s + 1.47·29-s − 1.66·31-s − 0.889·32-s − 0.822·33-s − 0.0706·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.019140234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019140234\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 288 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 122 T + p^{3} T^{2} \) |
| 43 | \( 1 - 188 T + p^{3} T^{2} \) |
| 47 | \( 1 + 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 338 T + p^{3} T^{2} \) |
| 59 | \( 1 - 100 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 - 84 T + p^{3} T^{2} \) |
| 71 | \( 1 + 328 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 - 330 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−14.26644976980438982787649456904, −12.87178185766002903819470036287, −11.73054756058014619978403436887, −10.75895408803862747110576516509, −9.427773009885794580275007107354, −8.438488393107358782091622156565, −7.07726748958248890231014750140, −5.29145824217140247968346475277, −4.23781748237289687757860095682, −1.19351150400239501566744730243,
1.19351150400239501566744730243, 4.23781748237289687757860095682, 5.29145824217140247968346475277, 7.07726748958248890231014750140, 8.438488393107358782091622156565, 9.427773009885794580275007107354, 10.75895408803862747110576516509, 11.73054756058014619978403436887, 12.87178185766002903819470036287, 14.26644976980438982787649456904
