| L(s) = 1 | − 2·2-s + 4·4-s − 2·7-s − 8·8-s − 70·11-s + 54·13-s + 4·14-s + 16·16-s + 22·17-s + 24·19-s + 140·22-s + 100·23-s − 108·26-s − 8·28-s − 216·29-s + 208·31-s − 32·32-s − 44·34-s − 254·37-s − 48·38-s + 206·41-s + 292·43-s − 280·44-s − 200·46-s + 320·47-s − 339·49-s + 216·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.107·7-s − 0.353·8-s − 1.91·11-s + 1.15·13-s + 0.0763·14-s + 1/4·16-s + 0.313·17-s + 0.289·19-s + 1.35·22-s + 0.906·23-s − 0.814·26-s − 0.0539·28-s − 1.38·29-s + 1.20·31-s − 0.176·32-s − 0.221·34-s − 1.12·37-s − 0.204·38-s + 0.784·41-s + 1.03·43-s − 0.959·44-s − 0.641·46-s + 0.993·47-s − 0.988·49-s + 0.576·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.173892477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.173892477\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 70 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 24 T + p^{3} T^{2} \) |
| 23 | \( 1 - 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 216 T + p^{3} T^{2} \) |
| 31 | \( 1 - 208 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 206 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 320 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 - 370 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 - 728 T + p^{3} T^{2} \) |
| 71 | \( 1 - 540 T + p^{3} T^{2} \) |
| 73 | \( 1 - 604 T + p^{3} T^{2} \) |
| 79 | \( 1 - 792 T + p^{3} T^{2} \) |
| 83 | \( 1 + 404 T + p^{3} T^{2} \) |
| 89 | \( 1 - 938 T + p^{3} T^{2} \) |
| 97 | \( 1 - 56 T + p^{3} T^{2} \) |
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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
−10.70019533635509439802594807433, −9.812384184432312460431727489770, −8.814924018148430389791483866640, −7.989335991850597822429643307538, −7.23648128947956715433138410292, −5.99401127430153314886023744070, −5.11447561237455028553443361840, −3.48745555153995450573136340352, −2.34642785422502337311390804850, −0.75690484674756074478906175810,
0.75690484674756074478906175810, 2.34642785422502337311390804850, 3.48745555153995450573136340352, 5.11447561237455028553443361840, 5.99401127430153314886023744070, 7.23648128947956715433138410292, 7.989335991850597822429643307538, 8.814924018148430389791483866640, 9.812384184432312460431727489770, 10.70019533635509439802594807433
