| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 5.16·11-s + 4.16·13-s + 14-s + 16-s + 5·17-s − 7.16·19-s − 5.16·22-s + 6.16·23-s − 4.16·26-s − 28-s + 4.16·29-s + 10.4·31-s − 32-s − 5·34-s + 2.83·37-s + 7.16·38-s − 10.3·41-s + 7·43-s + 5.16·44-s − 6.16·46-s − 2·47-s + 49-s + 4.16·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.377·7-s − 0.353·8-s + 1.55·11-s + 1.15·13-s + 0.267·14-s + 0.250·16-s + 1.21·17-s − 1.64·19-s − 1.10·22-s + 1.28·23-s − 0.816·26-s − 0.188·28-s + 0.772·29-s + 1.88·31-s − 0.176·32-s − 0.857·34-s + 0.466·37-s + 1.16·38-s − 1.61·41-s + 1.06·43-s + 0.778·44-s − 0.908·46-s − 0.291·47-s + 0.142·49-s + 0.577·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930382475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930382475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 7.16T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 2.67T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 + 2.67T + 89T^{2} \) |
| 97 | \( 1 + 5.16T + 97T^{2} \) |
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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
−7.83731308506799140536824553872, −6.86415138530147717806438025455, −6.43415733256265357033133317026, −6.03911324460189041572831435014, −4.91984699178279064789370734755, −4.04168262799334491690291267136, −3.41798970470322404365022221665, −2.56442670435538539188394207030, −1.39387882222477001709234003344, −0.847183912741118160516919371180,
0.847183912741118160516919371180, 1.39387882222477001709234003344, 2.56442670435538539188394207030, 3.41798970470322404365022221665, 4.04168262799334491690291267136, 4.91984699178279064789370734755, 6.03911324460189041572831435014, 6.43415733256265357033133317026, 6.86415138530147717806438025455, 7.83731308506799140536824553872
