L(s) = 1 | + (−6.41 − 11.1i)5-s + (6.32 + 17.4i)7-s + (−18.4 + 31.8i)11-s + 87.1·13-s + (51.2 − 88.8i)17-s + (−47.9 − 82.9i)19-s + (−48 − 83.1i)23-s + (−19.7 + 34.1i)25-s + 212.·29-s + (79.6 − 137. i)31-s + (152. − 181. i)35-s + (−64.3 − 111. i)37-s + 298.·41-s − 33.3·43-s + (135. + 234. i)47-s + ⋯ |
L(s) = 1 | + (−0.573 − 0.993i)5-s + (0.341 + 0.939i)7-s + (−0.504 + 0.874i)11-s + 1.85·13-s + (0.731 − 1.26i)17-s + (−0.578 − 1.00i)19-s + (−0.435 − 0.753i)23-s + (−0.157 + 0.273i)25-s + 1.35·29-s + (0.461 − 0.799i)31-s + (0.737 − 0.878i)35-s + (−0.285 − 0.495i)37-s + 1.13·41-s − 0.118·43-s + (0.420 + 0.728i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.684658976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684658976\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-6.32 - 17.4i)T \) |
good | 5 | \( 1 + (6.41 + 11.1i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (18.4 - 31.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-51.2 + 88.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (47.9 + 82.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (48 + 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-79.6 + 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (64.3 + 111. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-135. - 234. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-224. + 388. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (334. - 578. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-121. - 211. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-167. + 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-459. + 795. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-68.1 - 118. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-80.9 - 140. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 182.T + 9.12e5T^{2} \) |
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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
−11.67681807776979663869300378809, −10.64356955305887291278174028973, −9.297742384700761443751485349507, −8.576915495538355368034919589583, −7.80675738459040371438924046372, −6.34372779517028820917989802959, −5.13693794376654035208045225140, −4.28055608045716457502284919448, −2.55433378406029226241405310717, −0.823594089859419767699424459504,
1.23197585453533349326073923669, 3.33795141277712125428586543473, 3.96416539822846191491645940977, 5.80870439502315264124942488377, 6.67541479118251028953386166663, 7.984195033317557187493964641075, 8.389670027136311284370071343911, 10.27739738029949189745095937104, 10.71193782678460116594906511881, 11.42087154898875919365523938707