| L(s) = 1 | − 4.67·2-s + 13.8·4-s + 11.9·5-s + 26.1·7-s − 27.1·8-s − 55.6·10-s + 3.24·11-s − 20.0·13-s − 122.·14-s + 16.4·16-s + 17·17-s + 57.3·19-s + 164.·20-s − 15.1·22-s − 77.0·23-s + 17.0·25-s + 93.6·26-s + 361.·28-s + 286.·29-s − 8.54·31-s + 140.·32-s − 79.4·34-s + 311.·35-s + 357.·37-s − 267.·38-s − 324.·40-s − 194.·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.72·4-s + 1.06·5-s + 1.41·7-s − 1.20·8-s − 1.76·10-s + 0.0889·11-s − 0.427·13-s − 2.32·14-s + 0.257·16-s + 0.242·17-s + 0.692·19-s + 1.84·20-s − 0.146·22-s − 0.698·23-s + 0.136·25-s + 0.706·26-s + 2.43·28-s + 1.83·29-s − 0.0495·31-s + 0.777·32-s − 0.400·34-s + 1.50·35-s + 1.59·37-s − 1.14·38-s − 1.28·40-s − 0.740·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.077215835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.077215835\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 4.67T + 8T^{2} \) |
| 5 | \( 1 - 11.9T + 125T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 11 | \( 1 - 3.24T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.0T + 2.19e3T^{2} \) |
| 19 | \( 1 - 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 77.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8.54T + 2.97e4T^{2} \) |
| 37 | \( 1 - 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 74.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 939.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 520.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 953.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 486.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 685.T + 9.12e5T^{2} \) |
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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
−12.06355655455183979112984206493, −11.17733506649139507468506665512, −10.17207865755284213327310921791, −9.526513657271677198750921965248, −8.384103942534636298713570385177, −7.66339150973919625829796199995, −6.33393508384993200101546082744, −4.92366682842881873154332931004, −2.30137169809090431660128705933, −1.19248705018502435910495260381,
1.19248705018502435910495260381, 2.30137169809090431660128705933, 4.92366682842881873154332931004, 6.33393508384993200101546082744, 7.66339150973919625829796199995, 8.384103942534636298713570385177, 9.526513657271677198750921965248, 10.17207865755284213327310921791, 11.17733506649139507468506665512, 12.06355655455183979112984206493
