| L(s) = 1 | + 0.0645·2-s − 2.75·3-s − 1.99·4-s − 5-s − 0.177·6-s + 0.602·7-s − 0.258·8-s + 4.56·9-s − 0.0645·10-s + 11-s + 5.49·12-s + 2.04·13-s + 0.0389·14-s + 2.75·15-s + 3.97·16-s − 3.76·17-s + 0.295·18-s + 7.17·19-s + 1.99·20-s − 1.65·21-s + 0.0645·22-s + 5.12·23-s + 0.709·24-s + 25-s + 0.132·26-s − 4.31·27-s − 1.20·28-s + ⋯ |
| L(s) = 1 | + 0.0456·2-s − 1.58·3-s − 0.997·4-s − 0.447·5-s − 0.0725·6-s + 0.227·7-s − 0.0912·8-s + 1.52·9-s − 0.0204·10-s + 0.301·11-s + 1.58·12-s + 0.567·13-s + 0.0103·14-s + 0.710·15-s + 0.993·16-s − 0.913·17-s + 0.0695·18-s + 1.64·19-s + 0.446·20-s − 0.361·21-s + 0.0137·22-s + 1.06·23-s + 0.144·24-s + 0.200·25-s + 0.0259·26-s − 0.830·27-s − 0.227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6305558637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6305558637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0645T + 2T^{2} \) |
| 3 | \( 1 + 2.75T + 3T^{2} \) |
| 7 | \( 1 - 0.602T + 7T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 8.95T + 31T^{2} \) |
| 37 | \( 1 - 0.982T + 37T^{2} \) |
| 41 | \( 1 - 0.595T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 8.91T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 - 1.23T + 71T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 + 7.59T + 97T^{2} \) |
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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
−8.561381195809017723197989355697, −7.49040607849201520237159067171, −7.00992759627482783577912908804, −5.98253301555855410879386382987, −5.40789639813975405277726880297, −4.84851078675497140735474274997, −4.07761734371770887514302341435, −3.30246626849518930796706414406, −1.47024283151745315669381556666, −0.53840042453192057513746946911,
0.53840042453192057513746946911, 1.47024283151745315669381556666, 3.30246626849518930796706414406, 4.07761734371770887514302341435, 4.84851078675497140735474274997, 5.40789639813975405277726880297, 5.98253301555855410879386382987, 7.00992759627482783577912908804, 7.49040607849201520237159067171, 8.561381195809017723197989355697
