| L(s) = 1 | − 0.885·5-s − 3.81·7-s − 52.3·11-s − 8.06·13-s + 17·17-s + 66.5·19-s + 180.·23-s − 124.·25-s + 41.2·29-s + 34.9·31-s + 3.38·35-s + 130.·37-s + 17.9·41-s − 277.·43-s + 463.·47-s − 328.·49-s + 329.·53-s + 46.3·55-s + 678.·59-s + 340.·61-s + 7.13·65-s − 15.3·67-s − 670.·71-s + 193.·73-s + 199.·77-s − 1.08e3·79-s − 865.·83-s + ⋯ |
| L(s) = 1 | − 0.0792·5-s − 0.206·7-s − 1.43·11-s − 0.171·13-s + 0.242·17-s + 0.803·19-s + 1.63·23-s − 0.993·25-s + 0.264·29-s + 0.202·31-s + 0.0163·35-s + 0.579·37-s + 0.0682·41-s − 0.984·43-s + 1.43·47-s − 0.957·49-s + 0.855·53-s + 0.113·55-s + 1.49·59-s + 0.714·61-s + 0.0136·65-s − 0.0280·67-s − 1.12·71-s + 0.310·73-s + 0.295·77-s − 1.53·79-s − 1.14·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 5 | \( 1 + 0.885T + 125T^{2} \) |
| 7 | \( 1 + 3.81T + 343T^{2} \) |
| 11 | \( 1 + 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.06T + 2.19e3T^{2} \) |
| 19 | \( 1 - 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 15.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 379.T + 9.12e5T^{2} \) |
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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
−8.156692857714328555346807174247, −7.44954134188309430363162956853, −6.82852381485564862327205971918, −5.64058041712624400052685883212, −5.22408539022502848534894457061, −4.21263880107606274110875833112, −3.11648611945639323801901008189, −2.48835879140957574137469342786, −1.12636125852212057358165046075, 0,
1.12636125852212057358165046075, 2.48835879140957574137469342786, 3.11648611945639323801901008189, 4.21263880107606274110875833112, 5.22408539022502848534894457061, 5.64058041712624400052685883212, 6.82852381485564862327205971918, 7.44954134188309430363162956853, 8.156692857714328555346807174247
