| L(s) = 1 | + 3-s − 3.71·7-s + 9-s + 3.31·11-s + 13-s + 1.55·17-s − 5.33·19-s − 3.71·21-s + 0.442·23-s + 27-s + 2.56·29-s + 0.613·31-s + 3.31·33-s − 0.257·37-s + 39-s − 10.6·41-s + 12.6·43-s + 7.44·47-s + 6.78·49-s + 1.55·51-s − 5.39·53-s − 5.33·57-s − 13.1·59-s − 5.27·61-s − 3.71·63-s + 10.5·67-s + 0.442·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.40·7-s + 0.333·9-s + 1.00·11-s + 0.277·13-s + 0.377·17-s − 1.22·19-s − 0.810·21-s + 0.0923·23-s + 0.192·27-s + 0.477·29-s + 0.110·31-s + 0.577·33-s − 0.0423·37-s + 0.160·39-s − 1.65·41-s + 1.93·43-s + 1.08·47-s + 0.968·49-s + 0.218·51-s − 0.741·53-s − 0.706·57-s − 1.71·59-s − 0.675·61-s − 0.467·63-s + 1.28·67-s + 0.0533·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.115392379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115392379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 23 | \( 1 - 0.442T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 - 0.613T + 31T^{2} \) |
| 37 | \( 1 + 0.257T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 7.44T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 0.311T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.898601872183510745471942063650, −7.07887586582490824219196118760, −6.37412077137627625529700486489, −6.15005817220053809787088867415, −4.95515973118666087053754639419, −4.03998480593778221413283022259, −3.54836061815997226057730920375, −2.79790544488079651684554345416, −1.87353633001064472785425594687, −0.69762297965155179851538211371,
0.69762297965155179851538211371, 1.87353633001064472785425594687, 2.79790544488079651684554345416, 3.54836061815997226057730920375, 4.03998480593778221413283022259, 4.95515973118666087053754639419, 6.15005817220053809787088867415, 6.37412077137627625529700486489, 7.07887586582490824219196118760, 7.898601872183510745471942063650
