L(s) = 1 | + (8.32 − 14.4i)5-s + (−35.8 − 62.1i)11-s + 65.3·13-s + (−45.2 − 78.3i)17-s + (81.9 − 141. i)19-s + (39.6 − 68.6i)23-s + (−76.1 − 131. i)25-s + 43.2·29-s + (67.8 + 117. i)31-s + (−135. + 234. i)37-s + 152.·41-s − 177.·43-s + (22.8 − 39.5i)47-s + (−79.2 − 137. i)53-s − 1.19e3·55-s + ⋯ |
L(s) = 1 | + (0.744 − 1.29i)5-s + (−0.983 − 1.70i)11-s + 1.39·13-s + (−0.645 − 1.11i)17-s + (0.989 − 1.71i)19-s + (0.359 − 0.622i)23-s + (−0.609 − 1.05i)25-s + 0.277·29-s + (0.392 + 0.680i)31-s + (−0.601 + 1.04i)37-s + 0.579·41-s − 0.630·43-s + (0.0708 − 0.122i)47-s + (−0.205 − 0.355i)53-s − 2.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.396637069\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.396637069\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-8.32 + 14.4i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (35.8 + 62.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 65.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + (45.2 + 78.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-81.9 + 141. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-39.6 + 68.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 43.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-67.8 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (135. - 234. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-22.8 + 39.5i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (79.2 + 137. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-195. - 339. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-275. + 477. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (229. + 397. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 486.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-287. - 497. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-334. + 578. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 76.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + (683. - 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 242.T + 9.12e5T^{2} \) |
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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
−8.711515946617368848284092813001, −8.155409794265827714652161864934, −6.87179728202739378112796930087, −6.08165825421533488208767306558, −5.14575114693431456198446453538, −4.87371989059854072415944435530, −3.37430947090744999009211927348, −2.56164683230250128621251687191, −1.06102513046464211457337966011, −0.55789240474903158975653750166,
1.55437087984917348918888907581, 2.24023447967234734150769185113, 3.33588708107699080915692894774, 4.17134539914322392419670630180, 5.48161957508674115040743900842, 6.05047192521871448633918206274, 6.89328197539328353295116220359, 7.59608569495487152775456952984, 8.372494616504279968180372741172, 9.550027745940560039856825295432