| L(s) = 1 | − 19.1·5-s + 35.1·7-s − 26·11-s − 13·13-s + 36.2·17-s − 95.5·19-s + 161.·23-s + 240.·25-s + 91.3·29-s − 266.·31-s − 671.·35-s − 149.·37-s + 77.8·41-s + 183.·43-s + 60.6·47-s + 890.·49-s − 281.·53-s + 496.·55-s + 542.·59-s + 65.0·61-s + 248.·65-s − 1.03e3·67-s − 1.04e3·71-s + 483.·73-s − 912.·77-s − 1.33e3·79-s − 812.·83-s + ⋯ |
| L(s) = 1 | − 1.70·5-s + 1.89·7-s − 0.712·11-s − 0.277·13-s + 0.516·17-s − 1.15·19-s + 1.46·23-s + 1.92·25-s + 0.585·29-s − 1.54·31-s − 3.24·35-s − 0.664·37-s + 0.296·41-s + 0.651·43-s + 0.188·47-s + 2.59·49-s − 0.729·53-s + 1.21·55-s + 1.19·59-s + 0.136·61-s + 0.474·65-s − 1.88·67-s − 1.74·71-s + 0.774·73-s − 1.35·77-s − 1.90·79-s − 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 19.1T + 125T^{2} \) |
| 7 | \( 1 - 35.1T + 343T^{2} \) |
| 11 | \( 1 + 26T + 1.33e3T^{2} \) |
| 17 | \( 1 - 36.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 281.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 65.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 483.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 812.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 936.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 954.T + 9.12e5T^{2} \) |
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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
−8.820955294584894446044598683688, −8.384785115454501425788467483153, −7.55744647543958016321580198840, −7.17897015450648382630927060803, −5.47080627681346773017448771972, −4.70028180179705277933182355936, −4.04345563900305051544364612323, −2.76979733851762320103661811110, −1.35481029576526497355415479567, 0,
1.35481029576526497355415479567, 2.76979733851762320103661811110, 4.04345563900305051544364612323, 4.70028180179705277933182355936, 5.47080627681346773017448771972, 7.17897015450648382630927060803, 7.55744647543958016321580198840, 8.384785115454501425788467483153, 8.820955294584894446044598683688
