| L(s) = 1 | − 3.30·3-s + 1.63·7-s + 7.89·9-s − 4.67·11-s − 4.75·13-s − 1.41·17-s + 19-s − 5.38·21-s − 1.96·23-s − 16.1·27-s + 6.85·29-s + 5.08·31-s + 15.4·33-s + 10.6·37-s + 15.6·39-s − 4.52·41-s − 7.83·43-s − 10.9·47-s − 4.34·49-s + 4.66·51-s − 1.55·53-s − 3.30·57-s − 6.81·59-s − 0.109·61-s + 12.8·63-s − 10.5·67-s + 6.48·69-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.616·7-s + 2.63·9-s − 1.40·11-s − 1.31·13-s − 0.343·17-s + 0.229·19-s − 1.17·21-s − 0.409·23-s − 3.11·27-s + 1.27·29-s + 0.912·31-s + 2.68·33-s + 1.74·37-s + 2.51·39-s − 0.706·41-s − 1.19·43-s − 1.59·47-s − 0.620·49-s + 0.653·51-s − 0.214·53-s − 0.437·57-s − 0.887·59-s − 0.0139·61-s + 1.62·63-s − 1.29·67-s + 0.780·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5629188911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5629188911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 23 | \( 1 + 1.96T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 4.52T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 59 | \( 1 + 6.81T + 59T^{2} \) |
| 61 | \( 1 + 0.109T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 0.519T + 73T^{2} \) |
| 79 | \( 1 - 0.840T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 7.44T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−8.120624589273801885028025419296, −7.76468598900464449070858832962, −6.80497102187186364818282763424, −6.27765077601011522401563841547, −5.32764072181234475506334312861, −4.85837984769410735562740723784, −4.46648786347518582615362076038, −2.86946963991985976446731019380, −1.74476405464845389362770201299, −0.47486822745821865085859932953,
0.47486822745821865085859932953, 1.74476405464845389362770201299, 2.86946963991985976446731019380, 4.46648786347518582615362076038, 4.85837984769410735562740723784, 5.32764072181234475506334312861, 6.27765077601011522401563841547, 6.80497102187186364818282763424, 7.76468598900464449070858832962, 8.120624589273801885028025419296
