| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s − 3·9-s + 13-s + 3·14-s + 16-s + 5·17-s − 3·18-s + 19-s + 5·23-s − 5·25-s + 26-s + 3·28-s + 6·29-s + 2·31-s + 32-s + 5·34-s − 3·36-s − 3·37-s + 38-s − 2·41-s − 4·43-s + 5·46-s − 3·47-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s − 9-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 1.21·17-s − 0.707·18-s + 0.229·19-s + 1.04·23-s − 25-s + 0.196·26-s + 0.566·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.857·34-s − 1/2·36-s − 0.493·37-s + 0.162·38-s − 0.312·41-s − 0.609·43-s + 0.737·46-s − 0.437·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.619617200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.619617200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.150826888021398637056191027414, −7.72737780605976847335768104497, −6.74235365507796335395375483918, −5.98549737981023170966784358835, −5.20063175510659320460958370717, −4.85797070429121047217403731490, −3.70378531285287165276704846728, −3.04818784138374872426426344016, −2.04963302379774021754690589325, −1.00140990877260903715198078558,
1.00140990877260903715198078558, 2.04963302379774021754690589325, 3.04818784138374872426426344016, 3.70378531285287165276704846728, 4.85797070429121047217403731490, 5.20063175510659320460958370717, 5.98549737981023170966784358835, 6.74235365507796335395375483918, 7.72737780605976847335768104497, 8.150826888021398637056191027414
