| L(s) = 1 | + 0.541·2-s − 1.70·4-s + 1.08·7-s − 2.00·8-s − 2.91·11-s + 0.966·13-s + 0.589·14-s + 2.32·16-s − 2.65·17-s − 1.96·19-s − 1.57·22-s + 6.83·23-s + 0.523·26-s − 1.85·28-s + 6.81·29-s + 2.48·31-s + 5.27·32-s − 1.44·34-s − 8.22·37-s − 1.06·38-s + 10.5·41-s − 7.72·43-s + 4.97·44-s + 3.70·46-s − 9.02·47-s − 5.81·49-s − 1.64·52-s + ⋯ |
| L(s) = 1 | + 0.383·2-s − 0.853·4-s + 0.411·7-s − 0.710·8-s − 0.878·11-s + 0.267·13-s + 0.157·14-s + 0.581·16-s − 0.645·17-s − 0.451·19-s − 0.336·22-s + 1.42·23-s + 0.102·26-s − 0.350·28-s + 1.26·29-s + 0.446·31-s + 0.932·32-s − 0.247·34-s − 1.35·37-s − 0.172·38-s + 1.65·41-s − 1.17·43-s + 0.749·44-s + 0.546·46-s − 1.31·47-s − 0.830·49-s − 0.228·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.541T + 2T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 0.966T + 13T^{2} \) |
| 17 | \( 1 + 2.65T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 - 6.83T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.72T + 43T^{2} \) |
| 47 | \( 1 + 9.02T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 + 1.89T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−8.078609259493730564146338402693, −6.94665308063321959839219257461, −6.33640112430338577537819122100, −5.32437033397196954646897125126, −4.91206705384337370226209436189, −4.25965782650374262767766154683, −3.28664962918379533990506222935, −2.56137435374770645159337215765, −1.25500201329305091461279767000, 0,
1.25500201329305091461279767000, 2.56137435374770645159337215765, 3.28664962918379533990506222935, 4.25965782650374262767766154683, 4.91206705384337370226209436189, 5.32437033397196954646897125126, 6.33640112430338577537819122100, 6.94665308063321959839219257461, 8.078609259493730564146338402693
