| 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
−10.42742159732335419909950751930, −9.285560660293003633864848810905, −8.261227152270461626443946015735, −7.83585590572544441062544354583, −6.45215667214553317431699458332, −5.72176214258874655195980911260, −4.25678487168865724890900424863, −2.92238207896602842104644539769, −1.57819120348360981361813177911, 0,
1.57819120348360981361813177911, 2.92238207896602842104644539769, 4.25678487168865724890900424863, 5.72176214258874655195980911260, 6.45215667214553317431699458332, 7.83585590572544441062544354583, 8.261227152270461626443946015735, 9.285560660293003633864848810905, 10.42742159732335419909950751930
