| L(s) = 1 | + 1.67·2-s + 0.806·4-s + 1.19·7-s − 1.99·8-s + 4.28·11-s + 2.09·13-s + 2·14-s − 4.96·16-s − 2.83·17-s − 0.0376·19-s + 7.18·22-s + 3.89·23-s + 3.50·26-s + 0.962·28-s + 3.48·29-s − 31-s − 4.31·32-s − 4.74·34-s + 10.4·37-s − 0.0630·38-s + 4.06·41-s − 7.66·43-s + 3.45·44-s + 6.53·46-s + 0.574·47-s − 5.57·49-s + 1.68·52-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.403·4-s + 0.451·7-s − 0.707·8-s + 1.29·11-s + 0.580·13-s + 0.534·14-s − 1.24·16-s − 0.686·17-s − 0.00862·19-s + 1.53·22-s + 0.813·23-s + 0.687·26-s + 0.181·28-s + 0.646·29-s − 0.179·31-s − 0.762·32-s − 0.813·34-s + 1.72·37-s − 0.0102·38-s + 0.634·41-s − 1.16·43-s + 0.520·44-s + 0.963·46-s + 0.0838·47-s − 0.796·49-s + 0.233·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.053546755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.053546755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 - 2.09T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 + 0.0376T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 - 0.574T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + 3.98T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 1.68T + 71T^{2} \) |
| 73 | \( 1 - 0.649T + 73T^{2} \) |
| 79 | \( 1 - 4.86T + 79T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
| 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
−7.955353775778887135724320944515, −6.88238529165278226843343696578, −6.42873126416991343917864316923, −5.82578643657712762250269158877, −4.88263768077247186926032582544, −4.44712415730881415381015502546, −3.72339074589286317801329685197, −3.01877363898489315443522374980, −1.99303347216600128496186578117, −0.887016352075377794085993163698,
0.887016352075377794085993163698, 1.99303347216600128496186578117, 3.01877363898489315443522374980, 3.72339074589286317801329685197, 4.44712415730881415381015502546, 4.88263768077247186926032582544, 5.82578643657712762250269158877, 6.42873126416991343917864316923, 6.88238529165278226843343696578, 7.955353775778887135724320944515
