| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 4.47·13-s − 2·15-s − 4.47·17-s + 6.47·19-s + 21-s + 23-s − 25-s − 27-s − 2·29-s − 6.47·31-s − 2·35-s − 10.9·37-s − 4.47·39-s − 6·41-s − 12.9·43-s + 2·45-s − 6.47·47-s + 49-s + 4.47·51-s + 6.94·53-s − 6.47·57-s − 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 0.333·9-s + 1.24·13-s − 0.516·15-s − 1.08·17-s + 1.48·19-s + 0.218·21-s + 0.208·23-s − 0.200·25-s − 0.192·27-s − 0.371·29-s − 1.16·31-s − 0.338·35-s − 1.79·37-s − 0.716·39-s − 0.937·41-s − 1.97·43-s + 0.298·45-s − 0.944·47-s + 0.142·49-s + 0.626·51-s + 0.953·53-s − 0.857·57-s − 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 6.94T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−7.26615920217516378551446534422, −6.71988533343412656614816219155, −6.14868435834373147367720778183, −5.39304371429006967902685907062, −5.00998104458034890151586188188, −3.77383605413945551638739512375, −3.28150549102265351807757463815, −2.00082617240561853780350895273, −1.37552881952907943045177718703, 0,
1.37552881952907943045177718703, 2.00082617240561853780350895273, 3.28150549102265351807757463815, 3.77383605413945551638739512375, 5.00998104458034890151586188188, 5.39304371429006967902685907062, 6.14868435834373147367720778183, 6.71988533343412656614816219155, 7.26615920217516378551446534422
