| L(s) = 1 | − 9.98·2-s − 22.0·3-s + 67.7·4-s + 219.·6-s + 253.·7-s − 357.·8-s + 241.·9-s + 148.·11-s − 1.49e3·12-s + 867.·13-s − 2.52e3·14-s + 1.39e3·16-s + 1.30e3·17-s − 2.41e3·18-s − 1.32e3·19-s − 5.57e3·21-s − 1.48e3·22-s − 581.·23-s + 7.86e3·24-s − 8.66e3·26-s + 22.2·27-s + 1.71e4·28-s + 6.09e3·29-s − 8.05e3·31-s − 2.55e3·32-s − 3.26e3·33-s − 1.30e4·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.41·3-s + 2.11·4-s + 2.49·6-s + 1.95·7-s − 1.97·8-s + 0.995·9-s + 0.369·11-s − 2.99·12-s + 1.42·13-s − 3.44·14-s + 1.36·16-s + 1.09·17-s − 1.75·18-s − 0.843·19-s − 2.75·21-s − 0.652·22-s − 0.229·23-s + 2.78·24-s − 2.51·26-s + 0.00586·27-s + 4.13·28-s + 1.34·29-s − 1.50·31-s − 0.440·32-s − 0.522·33-s − 1.93·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9705015881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9705015881\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 + 9.98T + 32T^{2} \) |
| 3 | \( 1 + 22.0T + 243T^{2} \) |
| 7 | \( 1 - 253.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 148.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 867.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 581.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.32e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.46e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.43e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.80e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 638.T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.29e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.10e4T + 8.58e9T^{2} \) |
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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
−8.938250736768823235481022721397, −8.430622182316149400214352294465, −7.69891922291208348372324468327, −6.81661406569999939028404383994, −5.93934026053084547983230028162, −5.19397284855378676583556499934, −4.00310145391538190745131673883, −2.09498315487697058073519699847, −1.20579165551741423885061855704, −0.74279977778456095685130140150,
0.74279977778456095685130140150, 1.20579165551741423885061855704, 2.09498315487697058073519699847, 4.00310145391538190745131673883, 5.19397284855378676583556499934, 5.93934026053084547983230028162, 6.81661406569999939028404383994, 7.69891922291208348372324468327, 8.430622182316149400214352294465, 8.938250736768823235481022721397
