| L(s) = 1 | + 3.23·3-s − 4.47·7-s + 7.47·9-s + 1.23·13-s + 2.47·17-s + 19-s − 14.4·21-s + 2·23-s + 14.4·27-s + 2·29-s − 10.4·31-s + 9.23·37-s + 4.00·39-s + 12.4·41-s + 8.47·43-s + 3.52·47-s + 13.0·49-s + 8.00·51-s + 6.76·53-s + 3.23·57-s − 1.52·59-s + 4.47·61-s − 33.4·63-s − 3.23·67-s + 6.47·69-s − 10.4·71-s − 2.47·73-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 1.69·7-s + 2.49·9-s + 0.342·13-s + 0.599·17-s + 0.229·19-s − 3.15·21-s + 0.417·23-s + 2.78·27-s + 0.371·29-s − 1.88·31-s + 1.51·37-s + 0.640·39-s + 1.94·41-s + 1.29·43-s + 0.514·47-s + 1.85·49-s + 1.12·51-s + 0.929·53-s + 0.428·57-s − 0.198·59-s + 0.572·61-s − 4.21·63-s − 0.395·67-s + 0.779·69-s − 1.24·71-s − 0.289·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.487384103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.487384103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 3.23T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.737132529417887643461415941965, −7.57100635326905424108841505511, −7.43013972223217889728740671024, −6.42731569685805583715308044787, −5.64820166643338746615772917825, −4.21102734814250230906045666758, −3.70274219270030229542696992057, −2.93914915846392435370854411108, −2.40574850740826454819031934254, −1.02210976716129033086178362821,
1.02210976716129033086178362821, 2.40574850740826454819031934254, 2.93914915846392435370854411108, 3.70274219270030229542696992057, 4.21102734814250230906045666758, 5.64820166643338746615772917825, 6.42731569685805583715308044787, 7.43013972223217889728740671024, 7.57100635326905424108841505511, 8.737132529417887643461415941965
