| L(s) = 1 | − 3·3-s − 0.100·5-s + 9·9-s + 43.9·11-s − 16.6·13-s + 0.301·15-s − 121.·17-s + 127.·19-s − 53.5·23-s − 124.·25-s − 27·27-s + 235.·29-s + 18.7·31-s − 131.·33-s − 191.·37-s + 49.9·39-s − 319.·41-s + 218.·43-s − 0.904·45-s − 401.·47-s + 364.·51-s + 643.·53-s − 4.41·55-s − 381.·57-s + 11.6·59-s − 12.2·61-s + 1.67·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.00898·5-s + 0.333·9-s + 1.20·11-s − 0.355·13-s + 0.00519·15-s − 1.73·17-s + 1.53·19-s − 0.485·23-s − 0.999·25-s − 0.192·27-s + 1.50·29-s + 0.108·31-s − 0.695·33-s − 0.852·37-s + 0.205·39-s − 1.21·41-s + 0.775·43-s − 0.00299·45-s − 1.24·47-s + 1.00·51-s + 1.66·53-s − 0.0108·55-s − 0.886·57-s + 0.0256·59-s − 0.0256·61-s + 0.00319·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.100T + 125T^{2} \) |
| 11 | \( 1 - 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 643.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 11.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 12.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 805.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 394.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−8.313848049333632306757232578255, −7.27021255194847001970482923685, −6.68594000746668801681336151486, −6.00381531244504878724641929374, −5.02228339916917045622774538424, −4.31672928450321038007033513879, −3.42279087325664384098749913940, −2.17500084842736339701338460257, −1.16128508964478700119890521665, 0,
1.16128508964478700119890521665, 2.17500084842736339701338460257, 3.42279087325664384098749913940, 4.31672928450321038007033513879, 5.02228339916917045622774538424, 6.00381531244504878724641929374, 6.68594000746668801681336151486, 7.27021255194847001970482923685, 8.313848049333632306757232578255
