| L(s) = 1 | + (−0.267 + 0.462i)2-s + (3.78 − 3.55i)3-s + (3.85 + 6.68i)4-s − 1.39·5-s + (0.633 + 2.70i)6-s + (16.9 + 7.37i)7-s − 8.39·8-s + (1.71 − 26.9i)9-s + (0.372 − 0.644i)10-s + 19.3·11-s + (38.3 + 11.5i)12-s + (4.05 − 7.03i)13-s + (−7.95 + 5.88i)14-s + (−5.27 + 4.95i)15-s + (−28.6 + 49.5i)16-s + (28.8 − 49.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.0944 + 0.163i)2-s + (0.729 − 0.684i)3-s + (0.482 + 0.835i)4-s − 0.124·5-s + (0.0430 + 0.183i)6-s + (0.917 + 0.398i)7-s − 0.371·8-s + (0.0634 − 0.997i)9-s + (0.0117 − 0.0203i)10-s + 0.531·11-s + (0.923 + 0.279i)12-s + (0.0866 − 0.150i)13-s + (−0.151 + 0.112i)14-s + (−0.0908 + 0.0852i)15-s + (−0.447 + 0.774i)16-s + (0.411 − 0.713i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.160i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.89875 + 0.153812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89875 + 0.153812i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.78 + 3.55i)T \) |
| 7 | \( 1 + (-16.9 - 7.37i)T \) |
| good | 2 | \( 1 + (0.267 - 0.462i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 1.39T + 125T^{2} \) |
| 11 | \( 1 - 19.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-4.05 + 7.03i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-28.8 + 49.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-35.7 - 61.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (78.1 + 135. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (55.5 + 96.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-26.2 - 45.4i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (201. - 349. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-89.7 - 155. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-46.4 + 80.3i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-214. + 371. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-194. - 337. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-176. + 305. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (431. + 746. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 377.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-183. + 318. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-154. + 268. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-110. - 191. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-712. - 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (288. + 500. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.47203236137600981819226810078, −13.45561707378121654134123277071, −12.02661028742643874719311672376, −11.68131017291532320675896781846, −9.567768173467347285981505407981, −8.140791786685649036696737600916, −7.69772509689559203476871221118, −6.13584289999975360469560330343, −3.76032225999774483815593337362, −2.05695482323859447749831888278,
1.85866567112149151352676467799, 3.96602991740257066364768859338, 5.52245237878349566974872416349, 7.36787023791368216227039420585, 8.713719642635778292412123170702, 9.956388514247871655551927356390, 10.82783333800169526174162397279, 11.88514986653240110565153271528, 13.84979224708252331387375077232, 14.43135366016734469405045307381
