| L(s) = 1 | + 2.33·2-s + 3.43·4-s + 4.15·5-s + 1.65·7-s + 3.33·8-s + 9.68·10-s − 3.57·11-s + 5.72·13-s + 3.84·14-s + 0.908·16-s + 0.565·17-s + 4.78·19-s + 14.2·20-s − 8.32·22-s + 1.56·23-s + 12.2·25-s + 13.3·26-s + 5.66·28-s − 1.53·29-s − 4.55·32-s + 1.31·34-s + 6.86·35-s − 2.46·37-s + 11.1·38-s + 13.8·40-s − 5.37·41-s + 2.70·43-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 1.71·4-s + 1.85·5-s + 0.624·7-s + 1.17·8-s + 3.06·10-s − 1.07·11-s + 1.58·13-s + 1.02·14-s + 0.227·16-s + 0.137·17-s + 1.09·19-s + 3.18·20-s − 1.77·22-s + 0.325·23-s + 2.45·25-s + 2.61·26-s + 1.07·28-s − 0.284·29-s − 0.804·32-s + 0.226·34-s + 1.15·35-s − 0.405·37-s + 1.80·38-s + 2.19·40-s − 0.839·41-s + 0.412·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.367280443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.367280443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 - 0.565T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 + 5.37T + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 - 0.768T + 47T^{2} \) |
| 53 | \( 1 + 9.50T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 7.16T + 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 0.417T + 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−7.49037848499032701975644271251, −6.72311223710591506192419073113, −5.98745674732052821971720672161, −5.66073818042034059225627820666, −5.10684286473960644805559510306, −4.51537550750262454295369769765, −3.34180640345874361159242991195, −2.89768048737033848605937146703, −1.93108082411092706582752555683, −1.34693492279485888637241735979,
1.34693492279485888637241735979, 1.93108082411092706582752555683, 2.89768048737033848605937146703, 3.34180640345874361159242991195, 4.51537550750262454295369769765, 5.10684286473960644805559510306, 5.66073818042034059225627820666, 5.98745674732052821971720672161, 6.72311223710591506192419073113, 7.49037848499032701975644271251
