| L(s) = 1 | + 2.78·2-s − 1.81·3-s + 5.78·4-s + 5-s − 5.06·6-s + 0.269·7-s + 10.5·8-s + 0.292·9-s + 2.78·10-s + 1.81·11-s − 10.4·12-s + 0.752·14-s − 1.81·15-s + 17.8·16-s − 0.737·17-s + 0.816·18-s − 2.17·19-s + 5.78·20-s − 0.489·21-s + 5.05·22-s − 2.75·23-s − 19.1·24-s + 25-s + 4.91·27-s + 1.55·28-s − 7.21·29-s − 5.06·30-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 1.04·3-s + 2.89·4-s + 0.447·5-s − 2.06·6-s + 0.101·7-s + 3.72·8-s + 0.0976·9-s + 0.882·10-s + 0.545·11-s − 3.02·12-s + 0.200·14-s − 0.468·15-s + 4.46·16-s − 0.178·17-s + 0.192·18-s − 0.500·19-s + 1.29·20-s − 0.106·21-s + 1.07·22-s − 0.573·23-s − 3.90·24-s + 0.200·25-s + 0.945·27-s + 0.294·28-s − 1.33·29-s − 0.924·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.204081481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.204081481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 - 0.269T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 17 | \( 1 + 0.737T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 + 0.121T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 0.692T + 73T^{2} \) |
| 79 | \( 1 - 6.65T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 + 6.57T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−10.75247653080002290542738049825, −9.770992167795327571675038056001, −8.164314854611086828637820481981, −6.98368277798421797120032676183, −6.27293495764723429531605260685, −5.77432744415878342797590056014, −4.89393150475619995549174458424, −4.13835848571599910249198236230, −2.92427651014009707607285991907, −1.67135769249287824518621299783,
1.67135769249287824518621299783, 2.92427651014009707607285991907, 4.13835848571599910249198236230, 4.89393150475619995549174458424, 5.77432744415878342797590056014, 6.27293495764723429531605260685, 6.98368277798421797120032676183, 8.164314854611086828637820481981, 9.770992167795327571675038056001, 10.75247653080002290542738049825
