| L(s) = 1 | − 8.88·2-s − 9.77·3-s + 46.9·4-s + 25·5-s + 86.8·6-s + 148.·7-s − 132.·8-s − 147.·9-s − 222.·10-s + 725.·11-s − 458.·12-s − 409.·13-s − 1.32e3·14-s − 244.·15-s − 323.·16-s − 1.82e3·17-s + 1.30e3·18-s + 1.08e3·19-s + 1.17e3·20-s − 1.45e3·21-s − 6.44e3·22-s − 529·23-s + 1.29e3·24-s + 625·25-s + 3.63e3·26-s + 3.81e3·27-s + 6.99e3·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s − 0.627·3-s + 1.46·4-s + 0.447·5-s + 0.984·6-s + 1.14·7-s − 0.732·8-s − 0.606·9-s − 0.702·10-s + 1.80·11-s − 0.919·12-s − 0.671·13-s − 1.80·14-s − 0.280·15-s − 0.315·16-s − 1.53·17-s + 0.952·18-s + 0.690·19-s + 0.655·20-s − 0.720·21-s − 2.83·22-s − 0.208·23-s + 0.459·24-s + 0.200·25-s + 1.05·26-s + 1.00·27-s + 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7956928795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7956928795\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 8.88T + 32T^{2} \) |
| 3 | \( 1 + 9.77T + 243T^{2} \) |
| 7 | \( 1 - 148.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 725.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 409.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.82e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.08e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 856.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.50e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.04e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.23e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−11.92209701152694488644855097908, −11.41490097512925363654984903293, −10.49032986094663879231557148182, −9.225032243275025489295001464178, −8.636946420906433039146131960814, −7.27354182411802884293408759278, −6.22141898177385820633208870397, −4.64887986146833642794491643277, −2.07581304571033395672063556741, −0.836506830287488842400384665966,
0.836506830287488842400384665966, 2.07581304571033395672063556741, 4.64887986146833642794491643277, 6.22141898177385820633208870397, 7.27354182411802884293408759278, 8.636946420906433039146131960814, 9.225032243275025489295001464178, 10.49032986094663879231557148182, 11.41490097512925363654984903293, 11.92209701152694488644855097908
