| L(s) = 1 | + 2·2-s + 4·4-s − 7-s + 8·8-s − 42·11-s − 67·13-s − 2·14-s + 16·16-s − 54·17-s − 115·19-s − 84·22-s + 162·23-s − 134·26-s − 4·28-s + 210·29-s − 193·31-s + 32·32-s − 108·34-s − 286·37-s − 230·38-s − 12·41-s + 263·43-s − 168·44-s + 324·46-s − 414·47-s − 342·49-s − 268·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.0539·7-s + 0.353·8-s − 1.15·11-s − 1.42·13-s − 0.0381·14-s + 1/4·16-s − 0.770·17-s − 1.38·19-s − 0.814·22-s + 1.46·23-s − 1.01·26-s − 0.0269·28-s + 1.34·29-s − 1.11·31-s + 0.176·32-s − 0.544·34-s − 1.27·37-s − 0.981·38-s − 0.0457·41-s + 0.932·43-s − 0.575·44-s + 1.03·46-s − 1.28·47-s − 0.997·49-s − 0.714·52-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 67 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 115 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 193 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 12 T + p^{3} T^{2} \) |
| 43 | \( 1 - 263 T + p^{3} T^{2} \) |
| 47 | \( 1 + 414 T + p^{3} T^{2} \) |
| 53 | \( 1 - 192 T + p^{3} T^{2} \) |
| 59 | \( 1 + 690 T + p^{3} T^{2} \) |
| 61 | \( 1 + 733 T + p^{3} T^{2} \) |
| 67 | \( 1 - 299 T + p^{3} T^{2} \) |
| 71 | \( 1 - 228 T + p^{3} T^{2} \) |
| 73 | \( 1 - 938 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 462 T + p^{3} T^{2} \) |
| 89 | \( 1 - 240 T + p^{3} T^{2} \) |
| 97 | \( 1 + 511 T + p^{3} T^{2} \) |
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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
−10.53590891099361381586125100776, −9.383983358771029752170594587040, −8.286512841298656316969519409231, −7.27006923798158736534749385096, −6.47133082170900758326102848284, −5.15201860683402424611559286257, −4.59835388450073658273450453813, −3.07047455545715100279299436304, −2.12234604163918774211384388930, 0,
2.12234604163918774211384388930, 3.07047455545715100279299436304, 4.59835388450073658273450453813, 5.15201860683402424611559286257, 6.47133082170900758326102848284, 7.27006923798158736534749385096, 8.286512841298656316969519409231, 9.383983358771029752170594587040, 10.53590891099361381586125100776
