| L(s) = 1 | − 1.29·2-s − 6.31·4-s − 20.4·5-s + 29.9·7-s + 18.5·8-s + 26.5·10-s + 22.8·11-s − 44.4·13-s − 38.9·14-s + 26.3·16-s + 13.0·17-s + 5.41·19-s + 128.·20-s − 29.7·22-s + 175.·23-s + 291.·25-s + 57.7·26-s − 189.·28-s − 165.·29-s − 155.·31-s − 182.·32-s − 17.0·34-s − 611.·35-s − 95.3·37-s − 7.03·38-s − 379.·40-s − 189.·41-s + ⋯ |
| L(s) = 1 | − 0.459·2-s − 0.788·4-s − 1.82·5-s + 1.61·7-s + 0.822·8-s + 0.838·10-s + 0.626·11-s − 0.948·13-s − 0.743·14-s + 0.411·16-s + 0.186·17-s + 0.0653·19-s + 1.44·20-s − 0.288·22-s + 1.58·23-s + 2.33·25-s + 0.435·26-s − 1.27·28-s − 1.06·29-s − 0.901·31-s − 1.01·32-s − 0.0858·34-s − 2.95·35-s − 0.423·37-s − 0.0300·38-s − 1.50·40-s − 0.723·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 + 1.29T + 8T^{2} \) |
| 5 | \( 1 + 20.4T + 125T^{2} \) |
| 7 | \( 1 - 29.9T + 343T^{2} \) |
| 11 | \( 1 - 22.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.41T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 95.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 189.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 37.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 559.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 82.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 640.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 509.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 792.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 612.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 237.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 418.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 113.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−10.66438164230856907807314071348, −9.215637809524393676795947671530, −8.555434596588898489667081963147, −7.66097719887066460234457917103, −7.29124007950085640673885937769, −5.07113089963086051500087371127, −4.55596103853211547216056477973, −3.51256224523793588938939565594, −1.35879314000065357666312578326, 0,
1.35879314000065357666312578326, 3.51256224523793588938939565594, 4.55596103853211547216056477973, 5.07113089963086051500087371127, 7.29124007950085640673885937769, 7.66097719887066460234457917103, 8.555434596588898489667081963147, 9.215637809524393676795947671530, 10.66438164230856907807314071348
