| L(s) = 1 | − 1.48·2-s − 0.806·3-s + 0.193·4-s − 5-s + 1.19·6-s + 2.67·8-s − 2.35·9-s + 1.48·10-s + 4.15·11-s − 0.156·12-s − 2.96·13-s + 0.806·15-s − 4.35·16-s − 5.50·17-s + 3.48·18-s + 3.19·19-s − 0.193·20-s − 6.15·22-s + 1.84·23-s − 2.15·24-s + 25-s + 4.38·26-s + 4.31·27-s − 29-s − 1.19·30-s + 4.80·31-s + 1.09·32-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.465·3-s + 0.0969·4-s − 0.447·5-s + 0.487·6-s + 0.945·8-s − 0.783·9-s + 0.468·10-s + 1.25·11-s − 0.0451·12-s − 0.821·13-s + 0.208·15-s − 1.08·16-s − 1.33·17-s + 0.820·18-s + 0.732·19-s − 0.0433·20-s − 1.31·22-s + 0.384·23-s − 0.440·24-s + 0.200·25-s + 0.860·26-s + 0.829·27-s − 0.185·29-s − 0.217·30-s + 0.863·31-s + 0.193·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4586861505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4586861505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 + 0.806T + 3T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 5.50T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 0.0303T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 + 1.27T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 4.93T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 - 1.38T + 97T^{2} \) |
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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
−8.092275360094761632203239007667, −7.27486733961194973718940379309, −6.75636173695311193559496704987, −6.01325154075514951564649969806, −4.94709771753851750178196703070, −4.52948114101990415023136264367, −3.57430356662902259225696961675, −2.53867280187488933798538628050, −1.44393377522837089334996124045, −0.43930095579971578481891079714,
0.43930095579971578481891079714, 1.44393377522837089334996124045, 2.53867280187488933798538628050, 3.57430356662902259225696961675, 4.52948114101990415023136264367, 4.94709771753851750178196703070, 6.01325154075514951564649969806, 6.75636173695311193559496704987, 7.27486733961194973718940379309, 8.092275360094761632203239007667
