| L(s) = 1 | + 2-s − 1.27·3-s + 4-s − 1.27·6-s + 7-s + 8-s − 1.37·9-s − 4.10·11-s − 1.27·12-s − 4.65·13-s + 14-s + 16-s + 17-s − 1.37·18-s + 7.78·19-s − 1.27·21-s − 4.10·22-s + 1.82·23-s − 1.27·24-s − 4.65·26-s + 5.57·27-s + 28-s − 0.829·29-s + 5.33·31-s + 32-s + 5.22·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.735·3-s + 0.5·4-s − 0.520·6-s + 0.377·7-s + 0.353·8-s − 0.459·9-s − 1.23·11-s − 0.367·12-s − 1.28·13-s + 0.267·14-s + 0.250·16-s + 0.242·17-s − 0.324·18-s + 1.78·19-s − 0.277·21-s − 0.874·22-s + 0.379·23-s − 0.260·24-s − 0.912·26-s + 1.07·27-s + 0.188·28-s − 0.154·29-s + 0.957·31-s + 0.176·32-s + 0.909·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 19 | \( 1 - 7.78T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 0.829T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 - 5.05T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 - 0.821T + 47T^{2} \) |
| 53 | \( 1 + 6.33T + 53T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 2.67T + 73T^{2} \) |
| 79 | \( 1 - 0.651T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61002327997399326496217852022, −7.02638924881246879546079952260, −6.05316887131844758451513826185, −5.38260042797107342811452495467, −5.06023067560789852508376491574, −4.35534608723974008666962209075, −2.91946991146009019501398217735, −2.77433843164235772823500132321, −1.32026032854132052637175305174, 0,
1.32026032854132052637175305174, 2.77433843164235772823500132321, 2.91946991146009019501398217735, 4.35534608723974008666962209075, 5.06023067560789852508376491574, 5.38260042797107342811452495467, 6.05316887131844758451513826185, 7.02638924881246879546079952260, 7.61002327997399326496217852022
