| L(s) = 1 | + 5-s + 5.23·7-s + 1.23·11-s + 0.763·13-s − 4.47·17-s + 19-s − 2.47·23-s + 25-s − 0.763·29-s − 8.94·31-s + 5.23·35-s + 3.23·37-s + 9.70·41-s + 5.23·43-s − 2.47·47-s + 20.4·49-s + 0.472·53-s + 1.23·55-s + 10.4·59-s + 4.47·61-s + 0.763·65-s + 1.52·67-s + 12.4·73-s + 6.47·77-s − 4·83-s − 4.47·85-s + 1.70·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.97·7-s + 0.372·11-s + 0.211·13-s − 1.08·17-s + 0.229·19-s − 0.515·23-s + 0.200·25-s − 0.141·29-s − 1.60·31-s + 0.885·35-s + 0.532·37-s + 1.51·41-s + 0.798·43-s − 0.360·47-s + 2.91·49-s + 0.0648·53-s + 0.166·55-s + 1.36·59-s + 0.572·61-s + 0.0947·65-s + 0.186·67-s + 1.45·73-s + 0.737·77-s − 0.439·83-s − 0.485·85-s + 0.181·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.029231967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.029231967\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 0.763T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 4.76T + 97T^{2} \) |
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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
−7.87990044780814652222378503137, −7.44690062104442250101359635298, −6.56366173577286990451969727169, −5.71511838170767434762902238401, −5.17005435552545685518930338771, −4.38084393641928110390797943220, −3.82515413829852694352333434621, −2.38412167292024674129098985164, −1.89776623332570026488386716960, −0.939480966635311620287737337731,
0.939480966635311620287737337731, 1.89776623332570026488386716960, 2.38412167292024674129098985164, 3.82515413829852694352333434621, 4.38084393641928110390797943220, 5.17005435552545685518930338771, 5.71511838170767434762902238401, 6.56366173577286990451969727169, 7.44690062104442250101359635298, 7.87990044780814652222378503137
