| L(s) = 1 | − 2.69·3-s − 1.42·5-s + 4.27·9-s − 3.27·11-s − 3.42·13-s + 3.84·15-s − 5.27·17-s + 19-s + 0.574·23-s − 2.96·25-s − 3.42·27-s − 0.122·29-s − 2.12·31-s + 8.81·33-s − 3.96·37-s + 9.23·39-s + 4.66·41-s − 11.9·43-s − 6.08·45-s − 3.11·47-s + 14.2·51-s − 6.08·53-s + 4.66·55-s − 2.69·57-s + 1.72·59-s − 12.2·61-s + 4.88·65-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 0.637·5-s + 1.42·9-s − 0.986·11-s − 0.950·13-s + 0.992·15-s − 1.27·17-s + 0.229·19-s + 0.119·23-s − 0.593·25-s − 0.659·27-s − 0.0227·29-s − 0.381·31-s + 1.53·33-s − 0.652·37-s + 1.47·39-s + 0.728·41-s − 1.81·43-s − 0.907·45-s − 0.454·47-s + 1.98·51-s − 0.836·53-s + 0.628·55-s − 0.357·57-s + 0.224·59-s − 1.56·61-s + 0.605·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1873312769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1873312769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 23 | \( 1 - 0.574T + 23T^{2} \) |
| 29 | \( 1 + 0.122T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 6.81T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.355522684003646212508625082941, −7.61816682896335726648521725943, −6.96105404906214256907110247291, −6.29461274622035733698704405128, −5.39432915415220208762407129109, −4.88330718055391867467655135037, −4.21607902051530209116242952510, −3.01430408480641481016702309557, −1.83069808231681772991463293395, −0.26098274583688246480145691657,
0.26098274583688246480145691657, 1.83069808231681772991463293395, 3.01430408480641481016702309557, 4.21607902051530209116242952510, 4.88330718055391867467655135037, 5.39432915415220208762407129109, 6.29461274622035733698704405128, 6.96105404906214256907110247291, 7.61816682896335726648521725943, 8.355522684003646212508625082941
