| L(s) = 1 | + 2.34·2-s + 1.14·3-s + 3.48·4-s + 1.34·5-s + 2.68·6-s + 3.48·8-s − 1.68·9-s + 3.14·10-s + 1.14·11-s + 4.00·12-s − 13-s + 1.53·15-s + 1.19·16-s − 5.83·17-s − 3.94·18-s + 3.34·19-s + 4.68·20-s + 2.68·22-s − 3.17·23-s + 4.00·24-s − 3.19·25-s − 2.34·26-s − 5.37·27-s + 10.4·29-s + 3.60·30-s − 1.63·31-s − 4.17·32-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 0.661·3-s + 1.74·4-s + 0.600·5-s + 1.09·6-s + 1.23·8-s − 0.561·9-s + 0.994·10-s + 0.345·11-s + 1.15·12-s − 0.277·13-s + 0.397·15-s + 0.299·16-s − 1.41·17-s − 0.930·18-s + 0.766·19-s + 1.04·20-s + 0.572·22-s − 0.662·23-s + 0.816·24-s − 0.639·25-s − 0.459·26-s − 1.03·27-s + 1.94·29-s + 0.658·30-s − 0.293·31-s − 0.738·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.517336931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.517336931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 - 0.292T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.782T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 - 1.53T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 0.882T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−10.88133094215117859094217182785, −9.697945126639058778520668004483, −8.873697915208596104311527575212, −7.78625288877059975270319363518, −6.59010937519673736239124424302, −5.97417129810271293235761021520, −4.94775821718714729055606283018, −4.02691821596254686839092015711, −2.91868243165336358882903193607, −2.13648582623965500768110420761,
2.13648582623965500768110420761, 2.91868243165336358882903193607, 4.02691821596254686839092015711, 4.94775821718714729055606283018, 5.97417129810271293235761021520, 6.59010937519673736239124424302, 7.78625288877059975270319363518, 8.873697915208596104311527575212, 9.697945126639058778520668004483, 10.88133094215117859094217182785
