| L(s) = 1 | + 1.30·2-s − 0.302·4-s + 7-s − 3·8-s + 3·11-s + 4.60·13-s + 1.30·14-s − 3.30·16-s + 2.60·17-s − 0.605·19-s + 3.90·22-s − 8.21·23-s + 6·26-s − 0.302·28-s + 0.394·29-s + 7.21·31-s + 1.69·32-s + 3.39·34-s + 10.2·37-s − 0.788·38-s + 2.39·43-s − 0.908·44-s − 10.6·46-s − 3.39·47-s + 49-s − 1.39·52-s + 11.2·53-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 0.151·4-s + 0.377·7-s − 1.06·8-s + 0.904·11-s + 1.27·13-s + 0.348·14-s − 0.825·16-s + 0.631·17-s − 0.138·19-s + 0.833·22-s − 1.71·23-s + 1.17·26-s − 0.0572·28-s + 0.0732·29-s + 1.29·31-s + 0.300·32-s + 0.582·34-s + 1.67·37-s − 0.127·38-s + 0.365·43-s − 0.136·44-s − 1.57·46-s − 0.495·47-s + 0.142·49-s − 0.193·52-s + 1.53·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.656225012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.656225012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 + 8.21T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 6.60T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.454702779453558603687495098362, −8.493121564571527519054078223001, −8.035371209682523015740046840102, −6.64305296539398744761913097100, −6.05318035986079963216235704218, −5.31607799296967689680114561723, −4.08574816439250349541748956655, −3.90131234534287410232399866259, −2.57144016263289016065000871477, −1.07921911161642979531968838501,
1.07921911161642979531968838501, 2.57144016263289016065000871477, 3.90131234534287410232399866259, 4.08574816439250349541748956655, 5.31607799296967689680114561723, 6.05318035986079963216235704218, 6.64305296539398744761913097100, 8.035371209682523015740046840102, 8.493121564571527519054078223001, 9.454702779453558603687495098362
