| L(s) = 1 | − 3.17·5-s − 0.108·7-s + 1.62·11-s − 5.77·13-s − 4.70·17-s − 5.48·19-s + 0.0951·23-s + 5.07·25-s + 4.83·29-s − 8.21·31-s + 0.343·35-s + 5.57·37-s − 3.02·41-s + 6.54·43-s + 3.97·47-s − 6.98·49-s − 5.77·53-s − 5.16·55-s − 3.26·59-s − 2.91·61-s + 18.3·65-s − 11.7·67-s + 7.35·71-s − 4.72·73-s − 0.176·77-s − 7.99·79-s + 9.73·83-s + ⋯ |
| L(s) = 1 | − 1.41·5-s − 0.0408·7-s + 0.491·11-s − 1.60·13-s − 1.14·17-s − 1.25·19-s + 0.0198·23-s + 1.01·25-s + 0.898·29-s − 1.47·31-s + 0.0580·35-s + 0.917·37-s − 0.472·41-s + 0.998·43-s + 0.580·47-s − 0.998·49-s − 0.793·53-s − 0.696·55-s − 0.425·59-s − 0.372·61-s + 2.27·65-s − 1.44·67-s + 0.872·71-s − 0.553·73-s − 0.0200·77-s − 0.899·79-s + 1.06·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5863172310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5863172310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 + 0.108T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 5.77T + 13T^{2} \) |
| 17 | \( 1 + 4.70T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 - 0.0951T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 + 8.21T + 31T^{2} \) |
| 37 | \( 1 - 5.57T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 - 6.54T + 43T^{2} \) |
| 47 | \( 1 - 3.97T + 47T^{2} \) |
| 53 | \( 1 + 5.77T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 + 2.91T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 7.35T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 7.99T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 9.42T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.87392569460081702679207563705, −7.50844661944072321033280756046, −6.76450030013063155783532234349, −6.12566395291062823891689545488, −4.85649901149796522851680534938, −4.47806631447810589867321822805, −3.78304763091155588580270024424, −2.81940663945687644707508110491, −1.94834915596956575223736161667, −0.38160747089650236130579793918,
0.38160747089650236130579793918, 1.94834915596956575223736161667, 2.81940663945687644707508110491, 3.78304763091155588580270024424, 4.47806631447810589867321822805, 4.85649901149796522851680534938, 6.12566395291062823891689545488, 6.76450030013063155783532234349, 7.50844661944072321033280756046, 7.87392569460081702679207563705
