| L(s) = 1 | + 0.114·2-s + 5.80·3-s − 7.98·4-s − 11.2·5-s + 0.664·6-s + 15.9·7-s − 1.82·8-s + 6.70·9-s − 1.28·10-s + 11·11-s − 46.3·12-s + 1.82·14-s − 65.1·15-s + 63.6·16-s − 26.9·17-s + 0.768·18-s − 45.6·19-s + 89.6·20-s + 92.6·21-s + 1.25·22-s + 154.·23-s − 10.6·24-s + 0.974·25-s − 117.·27-s − 127.·28-s + 23.2·29-s − 7.45·30-s + ⋯ |
| L(s) = 1 | + 0.0404·2-s + 1.11·3-s − 0.998·4-s − 1.00·5-s + 0.0452·6-s + 0.862·7-s − 0.0808·8-s + 0.248·9-s − 0.0406·10-s + 0.301·11-s − 1.11·12-s + 0.0348·14-s − 1.12·15-s + 0.995·16-s − 0.384·17-s + 0.0100·18-s − 0.551·19-s + 1.00·20-s + 0.963·21-s + 0.0122·22-s + 1.40·23-s − 0.0903·24-s + 0.00779·25-s − 0.839·27-s − 0.860·28-s + 0.148·29-s − 0.0453·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.114T + 8T^{2} \) |
| 3 | \( 1 - 5.80T + 27T^{2} \) |
| 5 | \( 1 + 11.2T + 125T^{2} \) |
| 7 | \( 1 - 15.9T + 343T^{2} \) |
| 17 | \( 1 + 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 334.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 96.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 78.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 370.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 92.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 708.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 543.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 486.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 962.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 858.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 496.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 479.T + 9.12e5T^{2} \) |
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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
−8.543702637922048882910189632120, −7.85219646825550986659577140219, −7.38088403847689099315850449792, −6.01248229963208889527081267687, −4.87824233350432333643835130561, −4.25014527245187703333643923640, −3.55948257772579472638733720481, −2.59015996863944385600154844297, −1.26558100711962728134772564060, 0,
1.26558100711962728134772564060, 2.59015996863944385600154844297, 3.55948257772579472638733720481, 4.25014527245187703333643923640, 4.87824233350432333643835130561, 6.01248229963208889527081267687, 7.38088403847689099315850449792, 7.85219646825550986659577140219, 8.543702637922048882910189632120
