| L(s) = 1 | + 5-s + 2.61·7-s + 6.39·11-s + 13-s + 0.615·17-s + 7.77·19-s − 2.61·23-s + 25-s − 7.00·29-s + 5.23·31-s + 2.61·35-s + 7.16·37-s − 7.16·41-s + 4·43-s + 1.23·47-s − 0.161·49-s + 13.6·53-s + 6.39·55-s − 0.391·61-s + 65-s − 14.0·67-s − 5.16·71-s + 0.993·73-s + 16.7·77-s − 14.3·79-s − 18.0·83-s + 0.615·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.988·7-s + 1.92·11-s + 0.277·13-s + 0.149·17-s + 1.78·19-s − 0.545·23-s + 0.200·25-s − 1.30·29-s + 0.939·31-s + 0.442·35-s + 1.17·37-s − 1.11·41-s + 0.609·43-s + 0.179·47-s − 0.0230·49-s + 1.87·53-s + 0.861·55-s − 0.0501·61-s + 0.124·65-s − 1.71·67-s − 0.612·71-s + 0.116·73-s + 1.90·77-s − 1.61·79-s − 1.97·83-s + 0.0667·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083683014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083683014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 17 | \( 1 - 0.615T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 0.391T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 - 0.993T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
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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
−8.384665919645739147557303813211, −7.50328401201518424574131474747, −6.95081247507355643078917388942, −5.97834159388555341701342232441, −5.52778262827567581930659220682, −4.48524509190232527954781336997, −3.89404716041836738850998593859, −2.89478982675418459793522225020, −1.64050586077519938584336285424, −1.14062991009885315502905514346,
1.14062991009885315502905514346, 1.64050586077519938584336285424, 2.89478982675418459793522225020, 3.89404716041836738850998593859, 4.48524509190232527954781336997, 5.52778262827567581930659220682, 5.97834159388555341701342232441, 6.95081247507355643078917388942, 7.50328401201518424574131474747, 8.384665919645739147557303813211
