| L(s) = 1 | − 0.543·3-s − 2.11·7-s − 2.70·9-s − 4.99·11-s + 6.32·13-s + 3.75·17-s + 1.90·19-s + 1.14·21-s + 9.35·23-s + 3.09·27-s + 7.22·29-s + 0.994·31-s + 2.71·33-s − 6.37·37-s − 3.43·39-s + 4.18·41-s + 1.45·43-s + 2.78·47-s − 2.52·49-s − 2.04·51-s + 1.52·53-s − 1.03·57-s + 1.47·59-s + 8.79·61-s + 5.72·63-s − 12.0·67-s − 5.08·69-s + ⋯ |
| L(s) = 1 | − 0.313·3-s − 0.799·7-s − 0.901·9-s − 1.50·11-s + 1.75·13-s + 0.911·17-s + 0.437·19-s + 0.250·21-s + 1.94·23-s + 0.596·27-s + 1.34·29-s + 0.178·31-s + 0.472·33-s − 1.04·37-s − 0.550·39-s + 0.653·41-s + 0.222·43-s + 0.406·47-s − 0.360·49-s − 0.285·51-s + 0.209·53-s − 0.137·57-s + 0.192·59-s + 1.12·61-s + 0.720·63-s − 1.47·67-s − 0.611·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.193092662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.193092662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.543T + 3T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 - 9.35T + 23T^{2} \) |
| 29 | \( 1 - 7.22T + 29T^{2} \) |
| 31 | \( 1 - 0.994T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 9.09T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 3.08T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−10.19388816888478715002195662425, −9.001903722517744142311909790216, −8.438624430237509876577644486088, −7.46551736755786102655952818709, −6.42504344208708348142551815768, −5.69358268310058173865848038870, −4.95829748245161908284484670251, −3.38963219773857748805981466592, −2.83927531314506005597144705385, −0.867530540567614798367836764457,
0.867530540567614798367836764457, 2.83927531314506005597144705385, 3.38963219773857748805981466592, 4.95829748245161908284484670251, 5.69358268310058173865848038870, 6.42504344208708348142551815768, 7.46551736755786102655952818709, 8.438624430237509876577644486088, 9.001903722517744142311909790216, 10.19388816888478715002195662425
