| L(s) = 1 | − 2-s − 3-s + 4-s − 1.69·5-s + 6-s + 7-s − 8-s + 9-s + 1.69·10-s − 3.15·11-s − 12-s − 14-s + 1.69·15-s + 16-s + 7.74·17-s − 18-s − 3.15·19-s − 1.69·20-s − 21-s + 3.15·22-s + 7.89·23-s + 24-s − 2.13·25-s − 27-s + 28-s − 4.96·29-s − 1.69·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.756·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.535·10-s − 0.952·11-s − 0.288·12-s − 0.267·14-s + 0.436·15-s + 0.250·16-s + 1.87·17-s − 0.235·18-s − 0.724·19-s − 0.378·20-s − 0.218·21-s + 0.673·22-s + 1.64·23-s + 0.204·24-s − 0.427·25-s − 0.192·27-s + 0.188·28-s − 0.921·29-s − 0.308·30-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 | \( 1 + 1.69T + 5T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 17 | \( 1 - 7.74T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 0.185T + 37T^{2} \) |
| 41 | \( 1 + 0.978T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 + 0.652T + 53T^{2} \) |
| 59 | \( 1 - 4.80T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 7.33T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 7.83T + 97T^{2} \) |
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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
−7.57906266530831703125009652815, −7.19242965601963721800079680797, −6.26278525834785857698685654699, −5.34585718189374726178337902101, −5.04079945633783407773676134756, −3.82691017239928745516885558691, −3.19584665499108527770992913117, −2.06064177521110510403348339417, −1.03355157172470497198115310348, 0,
1.03355157172470497198115310348, 2.06064177521110510403348339417, 3.19584665499108527770992913117, 3.82691017239928745516885558691, 5.04079945633783407773676134756, 5.34585718189374726178337902101, 6.26278525834785857698685654699, 7.19242965601963721800079680797, 7.57906266530831703125009652815
