| L(s) = 1 | + 3-s + 2.35·5-s + 2.49·7-s + 9-s + 0.289·11-s + 4.02·13-s + 2.35·15-s + 6.97·17-s − 1.67·19-s + 2.49·21-s + 7.97·23-s + 0.537·25-s + 27-s + 7.86·29-s + 5.66·31-s + 0.289·33-s + 5.86·35-s − 9.36·37-s + 4.02·39-s + 6.84·41-s − 8.25·43-s + 2.35·45-s − 8.89·47-s − 0.790·49-s + 6.97·51-s − 13.6·53-s + 0.681·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.05·5-s + 0.941·7-s + 0.333·9-s + 0.0872·11-s + 1.11·13-s + 0.607·15-s + 1.69·17-s − 0.383·19-s + 0.543·21-s + 1.66·23-s + 0.107·25-s + 0.192·27-s + 1.46·29-s + 1.01·31-s + 0.0503·33-s + 0.991·35-s − 1.53·37-s + 0.644·39-s + 1.06·41-s − 1.25·43-s + 0.350·45-s − 1.29·47-s − 0.112·49-s + 0.976·51-s − 1.87·53-s + 0.0918·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.245814904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.245814904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 2.35T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 0.289T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 - 6.84T + 41T^{2} \) |
| 43 | \( 1 + 8.25T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 9.97T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 8.16T + 67T^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 - 2.19T + 89T^{2} \) |
| 97 | \( 1 - 8.42T + 97T^{2} \) |
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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
−8.178939985762905998979512942290, −7.53564978818428755821625317088, −6.51734119783577288280621915273, −6.05854323504796151457165605262, −5.06812290692207888921152319263, −4.65559765830651966102865812174, −3.37595825802831320283679318860, −2.89777586098963581711575693620, −1.57504643157253472520699477638, −1.30410361691749226604414036212,
1.30410361691749226604414036212, 1.57504643157253472520699477638, 2.89777586098963581711575693620, 3.37595825802831320283679318860, 4.65559765830651966102865812174, 5.06812290692207888921152319263, 6.05854323504796151457165605262, 6.51734119783577288280621915273, 7.53564978818428755821625317088, 8.178939985762905998979512942290
