| L(s) = 1 | + 2.04·3-s − 0.163·5-s + 3.50·7-s + 1.18·9-s − 2.69·13-s − 0.335·15-s + 7.57·17-s + 4.61·19-s + 7.16·21-s + 0.706·23-s − 4.97·25-s − 3.70·27-s + 1.02·29-s − 5.39·31-s − 0.573·35-s + 11.5·37-s − 5.51·39-s + 0.280·41-s + 4.31·43-s − 0.194·45-s − 5.82·47-s + 5.25·49-s + 15.5·51-s + 9.87·53-s + 9.45·57-s + 3.09·59-s − 6.93·61-s + ⋯ |
| L(s) = 1 | + 1.18·3-s − 0.0732·5-s + 1.32·7-s + 0.396·9-s − 0.747·13-s − 0.0865·15-s + 1.83·17-s + 1.05·19-s + 1.56·21-s + 0.147·23-s − 0.994·25-s − 0.713·27-s + 0.191·29-s − 0.969·31-s − 0.0969·35-s + 1.90·37-s − 0.882·39-s + 0.0438·41-s + 0.657·43-s − 0.0290·45-s − 0.850·47-s + 0.751·49-s + 2.17·51-s + 1.35·53-s + 1.25·57-s + 0.402·59-s − 0.887·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.092179731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092179731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 + 0.163T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 7.57T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 - 0.706T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 + 5.39T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 0.280T + 41T^{2} \) |
| 43 | \( 1 - 4.31T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 - 9.87T + 53T^{2} \) |
| 59 | \( 1 - 3.09T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 - 4.70T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−9.255412632096169883997497117505, −8.167786562522512224681200743672, −7.78487293444204153358534125159, −7.34927148444945221696381934045, −5.79556690961774165079391688764, −5.15826309875877454726844209689, −4.11812793405040925842758448085, −3.21849600833024890890885982120, −2.31850101459127591054879767248, −1.25448862496615872642083988921,
1.25448862496615872642083988921, 2.31850101459127591054879767248, 3.21849600833024890890885982120, 4.11812793405040925842758448085, 5.15826309875877454726844209689, 5.79556690961774165079391688764, 7.34927148444945221696381934045, 7.78487293444204153358534125159, 8.167786562522512224681200743672, 9.255412632096169883997497117505
