| L(s) = 1 | − 25.7·3-s − 109.·5-s − 177.·7-s + 420.·9-s + 228.·11-s − 476.·13-s + 2.81e3·15-s − 1.86e3·17-s − 361·19-s + 4.56e3·21-s + 1.96e3·23-s + 8.80e3·25-s − 4.57e3·27-s − 4.20e3·29-s − 3.84e3·31-s − 5.87e3·33-s + 1.93e4·35-s − 1.54e4·37-s + 1.22e4·39-s + 1.92e3·41-s − 7.33e3·43-s − 4.59e4·45-s + 5.08e3·47-s + 1.46e4·49-s + 4.81e4·51-s − 1.26e4·53-s − 2.49e4·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s − 1.95·5-s − 1.36·7-s + 1.73·9-s + 0.568·11-s − 0.781·13-s + 3.22·15-s − 1.56·17-s − 0.229·19-s + 2.26·21-s + 0.774·23-s + 2.81·25-s − 1.20·27-s − 0.927·29-s − 0.718·31-s − 0.938·33-s + 2.67·35-s − 1.85·37-s + 1.29·39-s + 0.179·41-s − 0.604·43-s − 3.38·45-s + 0.335·47-s + 0.870·49-s + 2.59·51-s − 0.616·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1211080215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1211080215\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 + 25.7T + 243T^{2} \) |
| 5 | \( 1 + 109.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 228.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 476.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.86e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.54e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.08e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.26e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.71e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.79e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−12.92494875313522705827480024417, −12.25407541420771425735302522872, −11.42560098733300591491520879725, −10.63518104743417938399057745356, −9.050439576922314909830268289978, −7.20521179131213236927060556530, −6.58451447593414980970943941913, −4.85605074952776546825579491217, −3.67637079863556572563571874213, −0.27305029903724779261781600937,
0.27305029903724779261781600937, 3.67637079863556572563571874213, 4.85605074952776546825579491217, 6.58451447593414980970943941913, 7.20521179131213236927060556530, 9.050439576922314909830268289978, 10.63518104743417938399057745356, 11.42560098733300591491520879725, 12.25407541420771425735302522872, 12.92494875313522705827480024417
