| L(s) = 1 | + 1.59·2-s + 0.553·4-s + 4.47·7-s − 2.31·8-s + 5.92·11-s − 5.46·13-s + 7.14·14-s − 4.80·16-s − 0.704·17-s + 2.56·19-s + 9.47·22-s + 4.43·23-s − 8.73·26-s + 2.47·28-s − 2.94·29-s + 2.35·31-s − 3.04·32-s − 1.12·34-s + 37-s + 4.09·38-s − 7.51·41-s + 12.2·43-s + 3.27·44-s + 7.09·46-s + 4.84·47-s + 12.9·49-s − 3.02·52-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.276·4-s + 1.69·7-s − 0.817·8-s + 1.78·11-s − 1.51·13-s + 1.90·14-s − 1.20·16-s − 0.170·17-s + 0.588·19-s + 2.01·22-s + 0.925·23-s − 1.71·26-s + 0.467·28-s − 0.547·29-s + 0.423·31-s − 0.538·32-s − 0.193·34-s + 0.164·37-s + 0.664·38-s − 1.17·41-s + 1.86·43-s + 0.494·44-s + 1.04·46-s + 0.706·47-s + 1.85·49-s − 0.419·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.537089875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.537089875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 0.704T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 41 | \( 1 + 7.51T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 + 0.897T + 53T^{2} \) |
| 59 | \( 1 + 5.55T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.59774772381768844782391733515, −7.06367889956783831458373305291, −6.30043872823968854634142612203, −5.39782864619704719293930905493, −4.96481768540500066915308903708, −4.35473015304961961847719323819, −3.80425469488745779492565376778, −2.76843871234839721195865489344, −1.92847277829028088231498009207, −0.926049292323505129632608368546,
0.926049292323505129632608368546, 1.92847277829028088231498009207, 2.76843871234839721195865489344, 3.80425469488745779492565376778, 4.35473015304961961847719323819, 4.96481768540500066915308903708, 5.39782864619704719293930905493, 6.30043872823968854634142612203, 7.06367889956783831458373305291, 7.59774772381768844782391733515
