| L(s) = 1 | + 1.65·3-s − 15.8·5-s − 7.01·7-s − 24.2·9-s + 35.1·11-s + 9.34·13-s − 26.2·15-s + 7.69·19-s − 11.6·21-s − 50.8·23-s + 125.·25-s − 84.9·27-s + 179.·29-s + 174.·31-s + 58.3·33-s + 110.·35-s + 227.·37-s + 15.4·39-s − 371.·41-s + 86.9·43-s + 383.·45-s + 376.·47-s − 293.·49-s − 33.4·53-s − 556.·55-s + 12.7·57-s + 323.·59-s + ⋯ |
| L(s) = 1 | + 0.319·3-s − 1.41·5-s − 0.378·7-s − 0.898·9-s + 0.964·11-s + 0.199·13-s − 0.451·15-s + 0.0928·19-s − 0.120·21-s − 0.460·23-s + 1.00·25-s − 0.605·27-s + 1.15·29-s + 1.01·31-s + 0.307·33-s + 0.535·35-s + 1.01·37-s + 0.0635·39-s − 1.41·41-s + 0.308·43-s + 1.27·45-s + 1.16·47-s − 0.856·49-s − 0.0866·53-s − 1.36·55-s + 0.0296·57-s + 0.714·59-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 | \( 1 - 1.65T + 27T^{2} \) |
| 5 | \( 1 + 15.8T + 125T^{2} \) |
| 7 | \( 1 + 7.01T + 343T^{2} \) |
| 11 | \( 1 - 35.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.34T + 2.19e3T^{2} \) |
| 19 | \( 1 - 7.69T + 6.85e3T^{2} \) |
| 23 | \( 1 + 50.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 86.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 376.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 33.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 323.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 685.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 981.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 658.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 215.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 452.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 549.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 140.T + 9.12e5T^{2} \) |
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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
−8.251754844845485185521351766499, −7.69993894312969450059609358803, −6.69567368960672267527972309028, −6.11860119545718680915604872158, −4.90481261597813931552874412319, −4.02668183370756640324008447470, −3.41896263101190706952504121289, −2.55850700466569266819768171784, −1.03272963928337302577104836018, 0,
1.03272963928337302577104836018, 2.55850700466569266819768171784, 3.41896263101190706952504121289, 4.02668183370756640324008447470, 4.90481261597813931552874412319, 6.11860119545718680915604872158, 6.69567368960672267527972309028, 7.69993894312969450059609358803, 8.251754844845485185521351766499
