| L(s) = 1 | + 0.150·2-s + 3.58·3-s − 7.97·4-s − 14.8·5-s + 0.539·6-s − 5.18·7-s − 2.40·8-s − 14.1·9-s − 2.23·10-s − 11·11-s − 28.5·12-s − 0.780·14-s − 53.3·15-s + 63.4·16-s − 1.86·17-s − 2.13·18-s + 122.·19-s + 118.·20-s − 18.5·21-s − 1.65·22-s + 193.·23-s − 8.61·24-s + 96.5·25-s − 147.·27-s + 41.3·28-s − 248.·29-s − 8.02·30-s + ⋯ |
| L(s) = 1 | + 0.0531·2-s + 0.689·3-s − 0.997·4-s − 1.33·5-s + 0.0366·6-s − 0.280·7-s − 0.106·8-s − 0.524·9-s − 0.0708·10-s − 0.301·11-s − 0.687·12-s − 0.0149·14-s − 0.917·15-s + 0.991·16-s − 0.0265·17-s − 0.0279·18-s + 1.47·19-s + 1.32·20-s − 0.193·21-s − 0.0160·22-s + 1.75·23-s − 0.0732·24-s + 0.772·25-s − 1.05·27-s + 0.279·28-s − 1.59·29-s − 0.0488·30-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 | \( 1 - 0.150T + 8T^{2} \) |
| 3 | \( 1 - 3.58T + 27T^{2} \) |
| 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 + 5.18T + 343T^{2} \) |
| 17 | \( 1 + 1.86T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 193.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 248.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 99.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 196.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 513.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 880.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 320.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 841.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 578.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 590.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−8.553631345548958097837724833298, −7.69389089037070840203539272983, −7.38917507208728191167355982054, −5.93888234818036040341765871100, −5.05330635490803336982070311907, −4.24040643451314184407873321467, −3.34823357033277465039841974061, −2.88879943243730484981015806158, −0.993950078795835006558220476988, 0,
0.993950078795835006558220476988, 2.88879943243730484981015806158, 3.34823357033277465039841974061, 4.24040643451314184407873321467, 5.05330635490803336982070311907, 5.93888234818036040341765871100, 7.38917507208728191167355982054, 7.69389089037070840203539272983, 8.553631345548958097837724833298
