| L(s) = 1 | − 1.11·2-s + 3.44·3-s − 0.753·4-s − 5-s − 3.85·6-s − 1.16·7-s + 3.07·8-s + 8.90·9-s + 1.11·10-s − 2.19·11-s − 2.59·12-s + 1.10·13-s + 1.30·14-s − 3.44·15-s − 1.92·16-s + 7.78·17-s − 9.94·18-s + 4.75·19-s + 0.753·20-s − 4.02·21-s + 2.44·22-s + 2.26·23-s + 10.6·24-s + 25-s − 1.22·26-s + 20.3·27-s + 0.879·28-s + ⋯ |
| L(s) = 1 | − 0.789·2-s + 1.99·3-s − 0.376·4-s − 0.447·5-s − 1.57·6-s − 0.441·7-s + 1.08·8-s + 2.96·9-s + 0.353·10-s − 0.660·11-s − 0.749·12-s + 0.305·13-s + 0.348·14-s − 0.890·15-s − 0.481·16-s + 1.88·17-s − 2.34·18-s + 1.08·19-s + 0.168·20-s − 0.878·21-s + 0.521·22-s + 0.473·23-s + 2.16·24-s + 0.200·25-s − 0.241·26-s + 3.91·27-s + 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542531168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542531168\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 - 3.44T + 3T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 - 7.78T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 + 6.01T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 0.949T + 47T^{2} \) |
| 53 | \( 1 - 8.03T + 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 2.59T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.60T + 83T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−10.51851595894414361524943626224, −9.874595278129134002062003927647, −9.217327350899455907717961789652, −8.445823446760631545196756063104, −7.67334378326403979973363369555, −7.26811067867902425845448333968, −5.14295398757228050423503924329, −3.75819356785121993097472253663, −3.10658656906357374549979497308, −1.42977022780599122831740594215,
1.42977022780599122831740594215, 3.10658656906357374549979497308, 3.75819356785121993097472253663, 5.14295398757228050423503924329, 7.26811067867902425845448333968, 7.67334378326403979973363369555, 8.445823446760631545196756063104, 9.217327350899455907717961789652, 9.874595278129134002062003927647, 10.51851595894414361524943626224
