| L(s) = 1 | + 2.03·3-s − 14.6·5-s + 9.92·7-s − 22.8·9-s + 26.9·11-s + 0.938·13-s − 29.7·15-s + 57.1·19-s + 20.1·21-s + 50.9·23-s + 89.1·25-s − 101.·27-s + 37.0·29-s − 263.·31-s + 54.8·33-s − 145.·35-s − 238.·37-s + 1.90·39-s + 163.·41-s + 325.·43-s + 334.·45-s + 471.·47-s − 244.·49-s + 142.·53-s − 394.·55-s + 116.·57-s − 282.·59-s + ⋯ |
| L(s) = 1 | + 0.391·3-s − 1.30·5-s + 0.535·7-s − 0.846·9-s + 0.739·11-s + 0.0200·13-s − 0.512·15-s + 0.689·19-s + 0.209·21-s + 0.462·23-s + 0.713·25-s − 0.722·27-s + 0.236·29-s − 1.52·31-s + 0.289·33-s − 0.701·35-s − 1.06·37-s + 0.00783·39-s + 0.621·41-s + 1.15·43-s + 1.10·45-s + 1.46·47-s − 0.712·49-s + 0.368·53-s − 0.968·55-s + 0.269·57-s − 0.622·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.03T + 27T^{2} \) |
| 5 | \( 1 + 14.6T + 125T^{2} \) |
| 7 | \( 1 - 9.92T + 343T^{2} \) |
| 11 | \( 1 - 26.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.938T + 2.19e3T^{2} \) |
| 19 | \( 1 - 57.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 37.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 163.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 142.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 282.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 461.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 411.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 729.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 156.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−8.249794519659808935598850219313, −7.57921366922341639902729952753, −6.99787328475702270992515135197, −5.83322268593228919953522864303, −5.02757103393424469992926715614, −3.98437878371178514290050248720, −3.49285012186897605676807962778, −2.43768235976317378655897890250, −1.14538811610505697642305710258, 0,
1.14538811610505697642305710258, 2.43768235976317378655897890250, 3.49285012186897605676807962778, 3.98437878371178514290050248720, 5.02757103393424469992926715614, 5.83322268593228919953522864303, 6.99787328475702270992515135197, 7.57921366922341639902729952753, 8.249794519659808935598850219313
