| L(s) = 1 | − 3·3-s + 10.6·5-s + 9·9-s − 6.65·11-s − 75.9·13-s − 31.9·15-s + 104.·17-s − 85.4·19-s − 68.6·23-s − 11.4·25-s − 27·27-s + 87.7·29-s − 62.7·31-s + 19.9·33-s + 42.2·37-s + 227.·39-s − 313.·41-s + 306.·43-s + 95.8·45-s − 215.·47-s − 312.·51-s + 525.·53-s − 70.8·55-s + 256.·57-s − 360.·59-s − 800.·61-s − 809.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.953·5-s + 0.333·9-s − 0.182·11-s − 1.62·13-s − 0.550·15-s + 1.48·17-s − 1.03·19-s − 0.622·23-s − 0.0917·25-s − 0.192·27-s + 0.562·29-s − 0.363·31-s + 0.105·33-s + 0.187·37-s + 0.935·39-s − 1.19·41-s + 1.08·43-s + 0.317·45-s − 0.667·47-s − 0.859·51-s + 1.36·53-s − 0.173·55-s + 0.595·57-s − 0.795·59-s − 1.68·61-s − 1.54·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 10.6T + 125T^{2} \) |
| 11 | \( 1 + 6.65T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 42.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 215.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 360.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 800.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 40.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 298.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 517.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−10.12714049391522965508650122502, −9.209285164303310246279296468776, −7.960258571737601823551325484289, −7.09350741436190570309337792608, −6.02142619955825502294064851900, −5.37026543394511362096076561078, −4.35988837380580326433221385717, −2.77817681564899282027104408059, −1.63549290811764027845480793228, 0,
1.63549290811764027845480793228, 2.77817681564899282027104408059, 4.35988837380580326433221385717, 5.37026543394511362096076561078, 6.02142619955825502294064851900, 7.09350741436190570309337792608, 7.960258571737601823551325484289, 9.209285164303310246279296468776, 10.12714049391522965508650122502
