| L(s) = 1 | − 2.91·5-s − 0.0334·7-s + 5.09·11-s + 2.03·13-s + 7.25·17-s + 5.42·19-s + 4.91·23-s + 3.51·25-s − 8.20·29-s − 10.1·31-s + 0.0975·35-s + 1.66·37-s + 41-s − 2.17·43-s + 9.69·47-s − 6.99·49-s + 6.88·53-s − 14.8·55-s − 5.83·59-s + 6.50·61-s − 5.93·65-s + 5.90·67-s − 10.7·71-s + 3.76·73-s − 0.170·77-s − 14.6·79-s − 3.59·83-s + ⋯ |
| L(s) = 1 | − 1.30·5-s − 0.0126·7-s + 1.53·11-s + 0.563·13-s + 1.75·17-s + 1.24·19-s + 1.02·23-s + 0.702·25-s − 1.52·29-s − 1.82·31-s + 0.0164·35-s + 0.273·37-s + 0.156·41-s − 0.331·43-s + 1.41·47-s − 0.999·49-s + 0.945·53-s − 2.00·55-s − 0.759·59-s + 0.832·61-s − 0.735·65-s + 0.720·67-s − 1.27·71-s + 0.440·73-s − 0.0193·77-s − 1.64·79-s − 0.394·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.879334942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.879334942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 2.91T + 5T^{2} \) |
| 7 | \( 1 + 0.0334T + 7T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 19 | \( 1 - 5.42T + 19T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 + 8.20T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 43 | \( 1 + 2.17T + 43T^{2} \) |
| 47 | \( 1 - 9.69T + 47T^{2} \) |
| 53 | \( 1 - 6.88T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 - 5.90T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 3.76T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 3.59T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 6.40T + 97T^{2} \) |
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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
−7.889900703645606005828425301640, −7.43143071226699828109691987424, −6.94332389095460583448108752278, −5.83911189235514225950226186206, −5.31981517454296603850549715177, −4.19394576270172048549748090995, −3.59244328703545067041057772437, −3.23700878150815761450774481422, −1.58936386430680815389799168344, −0.78799036300740047837642333936,
0.78799036300740047837642333936, 1.58936386430680815389799168344, 3.23700878150815761450774481422, 3.59244328703545067041057772437, 4.19394576270172048549748090995, 5.31981517454296603850549715177, 5.83911189235514225950226186206, 6.94332389095460583448108752278, 7.43143071226699828109691987424, 7.889900703645606005828425301640
