L(s) = 1 | + (−3.41 − 5.91i)5-s + (−14.9 + 10.9i)7-s + (50.5 + 29.1i)11-s − 38.5i·13-s + (−16.1 + 27.9i)17-s + (107. − 62.2i)19-s + (174. − 100. i)23-s + (39.2 − 67.9i)25-s + 104. i·29-s + (−240. − 138. i)31-s + (115. + 50.9i)35-s + (23.8 + 41.2i)37-s + 387.·41-s + 272.·43-s + (81.5 + 141. i)47-s + ⋯ |
L(s) = 1 | + (−0.305 − 0.528i)5-s + (−0.806 + 0.591i)7-s + (1.38 + 0.799i)11-s − 0.822i·13-s + (−0.230 + 0.398i)17-s + (1.30 − 0.751i)19-s + (1.57 − 0.911i)23-s + (0.313 − 0.543i)25-s + 0.668i·29-s + (−1.39 − 0.805i)31-s + (0.558 + 0.245i)35-s + (0.105 + 0.183i)37-s + 1.47·41-s + 0.966·43-s + (0.253 + 0.438i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.666221183\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666221183\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (14.9 - 10.9i)T \) |
good | 5 | \( 1 + (3.41 + 5.91i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-50.5 - 29.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (16.1 - 27.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-107. + 62.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-174. + 100. i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 104. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (240. + 138. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-23.8 - 41.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-81.5 - 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-313. - 181. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-105. + 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (202. - 117. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (262. - 454. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-465. - 268. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (362. + 628. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (430. + 744. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 978. iT - 9.12e5T^{2} \) |
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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
−11.70055678877625669734481137413, −10.60682316679174346494775363881, −9.228791300215412062260970000426, −9.042783842720600062790903922772, −7.50677821620670521424373749322, −6.55564567245134980592387787616, −5.34840089039768394917941588316, −4.13387034536692765983983993175, −2.78238634345708709309443282419, −0.889311145770136357043106808631,
1.09139915643883468909744306563, 3.20104354176271225602730418780, 3.95875061201793868481170937090, 5.65888245912891816604788517676, 6.85121102820222006137621474497, 7.38106055876219546081992369154, 9.064901253227257415738199776501, 9.522440744612369481487378924519, 10.94866266387134251321788121173, 11.46036079102081671753507003473