| L(s) = 1 | − 2-s − 3-s + 4-s + 2.56·5-s + 6-s − 7-s − 8-s + 9-s − 2.56·10-s − 1.56·11-s − 12-s + 14-s − 2.56·15-s + 16-s − 8.12·17-s − 18-s − 1.56·19-s + 2.56·20-s + 21-s + 1.56·22-s + 7.12·23-s + 24-s + 1.56·25-s − 27-s − 28-s + 2.12·29-s + 2.56·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.14·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.810·10-s − 0.470·11-s − 0.288·12-s + 0.267·14-s − 0.661·15-s + 0.250·16-s − 1.97·17-s − 0.235·18-s − 0.358·19-s + 0.572·20-s + 0.218·21-s + 0.332·22-s + 1.48·23-s + 0.204·24-s + 0.312·25-s − 0.192·27-s − 0.188·28-s + 0.394·29-s + 0.467·30-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 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 + 5.24T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 0.876T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.56T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 - 9.12T + 97T^{2} \) |
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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
−7.44361657808607727244718244351, −6.74081340930618384109669133126, −6.39780142324244468680838256697, −5.59096816164183695024004031050, −4.94622864617803131865073911221, −4.04388312798977435995292071836, −2.73116703306235969653789987380, −2.23352502003531102056837657603, −1.18249847667525229847670280195, 0,
1.18249847667525229847670280195, 2.23352502003531102056837657603, 2.73116703306235969653789987380, 4.04388312798977435995292071836, 4.94622864617803131865073911221, 5.59096816164183695024004031050, 6.39780142324244468680838256697, 6.74081340930618384109669133126, 7.44361657808607727244718244351
