| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (4.43 + 0.942i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (−4.08 + 4.53i)11-s + (0.913 − 0.406i)12-s + (3.67 + 1.63i)13-s + (3.03 + 3.36i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−2.65 − 2.94i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.0603 − 0.574i)3-s + (0.154 + 0.475i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (1.67 + 0.356i)7-s + (−0.109 + 0.336i)8-s + (−0.326 + 0.0693i)9-s + (−0.0330 + 0.314i)10-s + (−1.23 + 1.36i)11-s + (0.263 − 0.117i)12-s + (1.01 + 0.453i)13-s + (0.810 + 0.899i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.643 − 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08896 + 1.27924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08896 + 1.27924i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-3.03 - 4.66i)T \) |
| good | 7 | \( 1 + (-4.43 - 0.942i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (4.08 - 4.53i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.67 - 1.63i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (2.65 + 2.94i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.753 + 0.335i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.537 - 1.65i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.86 + 4.26i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.95 + 5.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.12 - 10.7i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-1.52 + 0.678i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 7.56i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 2.40i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.936 + 8.91i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 8.73T + 61T^{2} \) |
| 67 | \( 1 + (-0.693 - 1.20i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.84 + 2.09i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (7.55 - 8.39i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (6.80 + 7.55i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.361 + 3.43i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.490 - 1.50i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.09 - 6.45i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.41668169199317201680536981210, −9.180935062561990271797857148393, −8.225812523904400116429039631926, −7.57262471298807490178428095349, −6.92948601821207499389800161480, −5.76502278270954997297510633040, −5.05757905087412356864900227804, −4.22370107056090310294421909072, −2.55210893434131821008413731945, −1.79799392239545941404736843630,
1.04855154929564926669836571914, 2.43924092218908106339934192061, 3.74281296170332825073723558548, 4.54380077392202722923168160632, 5.51493280276747765042807450650, 5.90609494434173761158112086924, 7.60047561354904100911732396931, 8.394513554098186508264230103125, 8.936459470080292611998007511504, 10.42896998489417203592228067965
