L(s) = 1 | + (0.0412 − 1.73i)3-s + (0.707 − 0.707i)7-s + (−2.99 − 0.142i)9-s − 1.76i·11-s + (−0.719 − 0.719i)13-s + (1.55 + 1.55i)17-s − 5.12i·19-s + (−1.19 − 1.25i)21-s + (1.47 − 1.47i)23-s + (−0.371 + 5.18i)27-s − 7.63·29-s − 0.104·31-s + (−3.05 − 0.0728i)33-s + (−0.0126 + 0.0126i)37-s + (−1.27 + 1.21i)39-s + ⋯ |
L(s) = 1 | + (0.0238 − 0.999i)3-s + (0.267 − 0.267i)7-s + (−0.998 − 0.0476i)9-s − 0.532i·11-s + (−0.199 − 0.199i)13-s + (0.376 + 0.376i)17-s − 1.17i·19-s + (−0.260 − 0.273i)21-s + (0.306 − 0.306i)23-s + (−0.0714 + 0.997i)27-s − 1.41·29-s − 0.0187·31-s + (−0.532 − 0.0126i)33-s + (−0.00207 + 0.00207i)37-s + (−0.204 + 0.194i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.053687655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053687655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0412 + 1.73i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + 1.76iT - 11T^{2} \) |
| 13 | \( 1 + (0.719 + 0.719i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.55 - 1.55i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.12iT - 19T^{2} \) |
| 23 | \( 1 + (-1.47 + 1.47i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 0.104T + 31T^{2} \) |
| 37 | \( 1 + (0.0126 - 0.0126i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.35iT - 41T^{2} \) |
| 43 | \( 1 + (2.66 + 2.66i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.98 - 3.98i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.44 - 5.44i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + (3.01 - 3.01i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.20iT - 71T^{2} \) |
| 73 | \( 1 + (7.10 + 7.10i)T + 73iT^{2} \) |
| 79 | \( 1 + 15.4iT - 79T^{2} \) |
| 83 | \( 1 + (3.76 - 3.76i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + (-6.41 + 6.41i)T - 97iT^{2} \) |
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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
−8.787950376826825842516356275332, −7.68665954239754480964417236508, −7.38874606845598766608632073094, −6.39779167043339849343603371873, −5.69480706674109206163920309576, −4.81422691441120719804560867251, −3.58856733033637739746204838845, −2.66067909231265519905561846503, −1.57790731116526137928310473593, −0.36004579937816279748019412998,
1.68778612268911507121627485298, 2.90067020390629221458652558089, 3.79942429638899053407157615855, 4.64941471223359576506564924588, 5.40270426477066574249145534520, 6.11445561788004390615074322974, 7.27278794655977459832295961202, 8.030424866488496470164084000994, 8.800430084739392569127522046494, 9.719177335372579281699176292560