| L(s) = 1 | + 2-s + 4-s + 4.34·5-s − 0.852·7-s + 8-s + 4.34·10-s + 4.30·11-s + 6.03·13-s − 0.852·14-s + 16-s + 0.821·17-s − 0.0588·19-s + 4.34·20-s + 4.30·22-s − 8.16·23-s + 13.9·25-s + 6.03·26-s − 0.852·28-s + 8.57·29-s + 5.01·31-s + 32-s + 0.821·34-s − 3.70·35-s + 0.264·37-s − 0.0588·38-s + 4.34·40-s − 7.70·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.94·5-s − 0.322·7-s + 0.353·8-s + 1.37·10-s + 1.29·11-s + 1.67·13-s − 0.227·14-s + 0.250·16-s + 0.199·17-s − 0.0134·19-s + 0.972·20-s + 0.918·22-s − 1.70·23-s + 2.78·25-s + 1.18·26-s − 0.161·28-s + 1.59·29-s + 0.900·31-s + 0.176·32-s + 0.140·34-s − 0.626·35-s + 0.0434·37-s − 0.00954·38-s + 0.687·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.633169285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.633169285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 353 | \( 1 - T \) |
| good | 5 | \( 1 - 4.34T + 5T^{2} \) |
| 7 | \( 1 + 0.852T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 6.03T + 13T^{2} \) |
| 17 | \( 1 - 0.821T + 17T^{2} \) |
| 19 | \( 1 + 0.0588T + 19T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 0.264T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 0.788T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 + 4.12T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 + 2.19T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.286061640867613629552036499027, −6.68650137618800396140081151693, −6.46727791581042128740159017022, −6.09226279153897049401591307418, −5.29626772467104760366672686986, −4.46971504967845003642967646895, −3.56155108168253109663130973497, −2.85678115855298243965965540711, −1.70094444547147849969243071169, −1.34133273472311438906095737826,
1.34133273472311438906095737826, 1.70094444547147849969243071169, 2.85678115855298243965965540711, 3.56155108168253109663130973497, 4.46971504967845003642967646895, 5.29626772467104760366672686986, 6.09226279153897049401591307418, 6.46727791581042128740159017022, 6.68650137618800396140081151693, 8.286061640867613629552036499027
