| L(s) = 1 | − 2.26·2-s + 3.43·3-s + 3.14·4-s − 7.78·6-s + 3.42·7-s − 2.58·8-s + 8.78·9-s + 2.42·11-s + 10.7·12-s − 1.02·13-s − 7.75·14-s − 0.411·16-s − 19.9·18-s − 2.06·19-s + 11.7·21-s − 5.49·22-s + 0.137·23-s − 8.88·24-s + 2.32·26-s + 19.8·27-s + 10.7·28-s − 5.43·29-s + 5.31·31-s + 6.11·32-s + 8.32·33-s + 27.5·36-s − 8.82·37-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.98·3-s + 1.57·4-s − 3.17·6-s + 1.29·7-s − 0.915·8-s + 2.92·9-s + 0.731·11-s + 3.11·12-s − 0.284·13-s − 2.07·14-s − 0.102·16-s − 4.69·18-s − 0.474·19-s + 2.56·21-s − 1.17·22-s + 0.0287·23-s − 1.81·24-s + 0.455·26-s + 3.82·27-s + 2.03·28-s − 1.01·29-s + 0.954·31-s + 1.08·32-s + 1.44·33-s + 4.59·36-s − 1.45·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.728930760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.728930760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 3.43T + 3T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 23 | \( 1 - 0.137T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 + 3.25T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 + 0.158T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 + 0.926T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.72T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 9.77T + 97T^{2} \) |
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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
−8.153340039908550620915670894869, −7.46271759314648236866395724614, −7.24432252983114609692970224172, −6.25204529336879161702053397359, −4.77449803992934497372785978444, −4.20984922175845850738452471142, −3.25997384258245844099866668849, −2.25355543636937870314310732100, −1.81855900205732940767082592990, −1.05219549257276344837031927663,
1.05219549257276344837031927663, 1.81855900205732940767082592990, 2.25355543636937870314310732100, 3.25997384258245844099866668849, 4.20984922175845850738452471142, 4.77449803992934497372785978444, 6.25204529336879161702053397359, 7.24432252983114609692970224172, 7.46271759314648236866395724614, 8.153340039908550620915670894869
