| L(s) = 1 | − 0.618·3-s − 4.19·5-s + 4.20·7-s − 2.61·9-s − 13-s + 2.58·15-s + 6.49·17-s − 6.80·19-s − 2.60·21-s + 4.79·23-s + 12.5·25-s + 3.47·27-s − 4.01·29-s − 2.87·31-s − 17.6·35-s − 4.47·37-s + 0.618·39-s − 9.40·41-s + 7.11·43-s + 10.9·45-s − 4.49·47-s + 10.7·49-s − 4.01·51-s + 1.91·53-s + 4.20·57-s + 4.52·59-s − 5.31·61-s + ⋯ |
| L(s) = 1 | − 0.356·3-s − 1.87·5-s + 1.59·7-s − 0.872·9-s − 0.277·13-s + 0.668·15-s + 1.57·17-s − 1.56·19-s − 0.567·21-s + 1.00·23-s + 2.51·25-s + 0.668·27-s − 0.745·29-s − 0.515·31-s − 2.98·35-s − 0.735·37-s + 0.0989·39-s − 1.46·41-s + 1.08·43-s + 1.63·45-s − 0.656·47-s + 1.52·49-s − 0.562·51-s + 0.262·53-s + 0.557·57-s + 0.589·59-s − 0.680·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9683211663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9683211663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 + 4.49T + 47T^{2} \) |
| 53 | \( 1 - 1.91T + 53T^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 + 5.31T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 - 8.78T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 19.5T + 97T^{2} \) |
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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
−8.021548764547721321219383639454, −7.50854260976658181224196900260, −6.86168354645510577238514299249, −5.71704115996062447224333669805, −5.04825705584452103454204109846, −4.50917735407918928200850548121, −3.69405622459000371753884373223, −2.95931208425641300333923599950, −1.68334873922192736525651439713, −0.52891825573098047839736742534,
0.52891825573098047839736742534, 1.68334873922192736525651439713, 2.95931208425641300333923599950, 3.69405622459000371753884373223, 4.50917735407918928200850548121, 5.04825705584452103454204109846, 5.71704115996062447224333669805, 6.86168354645510577238514299249, 7.50854260976658181224196900260, 8.021548764547721321219383639454
