| L(s) = 1 | − 4.76·3-s − 6.38·5-s + 20.9·7-s − 4.33·9-s + 22.2·11-s − 31.8·13-s + 30.4·15-s + 80.6·19-s − 99.7·21-s − 5.09·23-s − 84.2·25-s + 149.·27-s − 212.·29-s + 26.3·31-s − 106.·33-s − 133.·35-s − 13.4·37-s + 151.·39-s − 318.·41-s + 241.·43-s + 27.7·45-s + 81.8·47-s + 96.4·49-s + 166.·53-s − 142.·55-s − 384.·57-s + 491.·59-s + ⋯ |
| L(s) = 1 | − 0.916·3-s − 0.571·5-s + 1.13·7-s − 0.160·9-s + 0.610·11-s − 0.680·13-s + 0.523·15-s + 0.974·19-s − 1.03·21-s − 0.0462·23-s − 0.673·25-s + 1.06·27-s − 1.36·29-s + 0.152·31-s − 0.559·33-s − 0.646·35-s − 0.0598·37-s + 0.623·39-s − 1.21·41-s + 0.856·43-s + 0.0917·45-s + 0.254·47-s + 0.281·49-s + 0.430·53-s − 0.348·55-s − 0.892·57-s + 1.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 4.76T + 27T^{2} \) |
| 5 | \( 1 + 6.38T + 125T^{2} \) |
| 7 | \( 1 - 20.9T + 343T^{2} \) |
| 11 | \( 1 - 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 80.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 5.09T + 1.21e4T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 26.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 13.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 241.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 81.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 166.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 491.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 385.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 893.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 598.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 354.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 939.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.125918437747628299571847092285, −7.51144870944731062764994808129, −6.75958802902900506648286009220, −5.69571122445003730195094348369, −5.19704898381577018048502841495, −4.37641030613983015931910568847, −3.47719421218236089787098973297, −2.15551512788397446279843101321, −1.06067857590809200419295988524, 0,
1.06067857590809200419295988524, 2.15551512788397446279843101321, 3.47719421218236089787098973297, 4.37641030613983015931910568847, 5.19704898381577018048502841495, 5.69571122445003730195094348369, 6.75958802902900506648286009220, 7.51144870944731062764994808129, 8.125918437747628299571847092285
