| L(s) = 1 | − 2-s − 3-s + 4-s + 3.04·5-s + 6-s + 7-s − 8-s + 9-s − 3.04·10-s − 2.13·11-s − 12-s − 14-s − 3.04·15-s + 16-s − 1.40·17-s − 18-s − 2.13·19-s + 3.04·20-s − 21-s + 2.13·22-s − 2.26·23-s + 24-s + 4.29·25-s − 27-s + 28-s − 2.58·29-s + 3.04·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.36·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.964·10-s − 0.644·11-s − 0.288·12-s − 0.267·14-s − 0.787·15-s + 0.250·16-s − 0.340·17-s − 0.235·18-s − 0.490·19-s + 0.681·20-s − 0.218·21-s + 0.455·22-s − 0.473·23-s + 0.204·24-s + 0.859·25-s − 0.192·27-s + 0.188·28-s − 0.479·29-s + 0.556·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 2.13T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 + 9.55T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 61 | \( 1 - 9.98T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4.21T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 - 5.10T + 97T^{2} \) |
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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
−7.52888200020319838398878184229, −6.86484543544051769676135209766, −6.19184261505219156836423564625, −5.58624743718854129833296972389, −5.02442155151442199181996127126, −4.04925207910072794797439878934, −2.77751272083382879590863362621, −2.05331140086636103197817411592, −1.33560986602549434645913016589, 0,
1.33560986602549434645913016589, 2.05331140086636103197817411592, 2.77751272083382879590863362621, 4.04925207910072794797439878934, 5.02442155151442199181996127126, 5.58624743718854129833296972389, 6.19184261505219156836423564625, 6.86484543544051769676135209766, 7.52888200020319838398878184229
