| L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.909 − 0.415i)3-s + (0.959 + 0.281i)4-s + (−2.09 + 0.792i)5-s + (−0.959 + 0.281i)6-s + (−0.378 + 0.588i)7-s + (−0.909 − 0.415i)8-s + (0.654 − 0.755i)9-s + (2.18 − 0.486i)10-s + (−0.745 − 5.18i)11-s + (0.989 − 0.142i)12-s + (1.92 + 2.99i)13-s + (0.457 − 0.528i)14-s + (−1.57 + 1.58i)15-s + (0.841 + 0.540i)16-s + (−0.765 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.100i)2-s + (0.525 − 0.239i)3-s + (0.479 + 0.140i)4-s + (−0.935 + 0.354i)5-s + (−0.391 + 0.115i)6-s + (−0.142 + 0.222i)7-s + (−0.321 − 0.146i)8-s + (0.218 − 0.251i)9-s + (0.690 − 0.153i)10-s + (−0.224 − 1.56i)11-s + (0.285 − 0.0410i)12-s + (0.534 + 0.832i)13-s + (0.122 − 0.141i)14-s + (−0.406 + 0.410i)15-s + (0.210 + 0.135i)16-s + (−0.185 − 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.555437 - 0.618840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.555437 - 0.618840i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.989 + 0.142i)T \) |
| 3 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (2.09 - 0.792i)T \) |
| 23 | \( 1 + (0.308 + 4.78i)T \) |
| good | 7 | \( 1 + (0.378 - 0.588i)T + (-2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.745 + 5.18i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 2.99i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.765 + 2.60i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (3.73 + 1.09i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.22 + 2.41i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.66 + 5.82i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (4.34 + 3.76i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.31 + 1.51i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.36 - 1.99i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 5.47iT - 47T^{2} \) |
| 53 | \( 1 + (1.05 - 1.64i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (9.17 - 5.89i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.71 + 12.5i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-3.44 - 0.495i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0479 - 0.333i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.65 + 12.4i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (3.98 - 2.55i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (12.2 + 10.6i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.75 - 8.22i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (5.51 - 4.78i)T + (13.8 - 96.0i)T^{2} \) |
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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
−10.31949210347229145346714224988, −9.055499804833808007754715725031, −8.518788909739312906338660095962, −7.926443635069747378796580534798, −6.77724071066080495114365132431, −6.19999338194197618340334965365, −4.45134272387735040144317043397, −3.35016104939503702275097425335, −2.43355868710672843142917510633, −0.53497467625634369192266224158,
1.51196139799478377140498685523, 3.05951936153958769031781583977, 4.15163065452570825223617159750, 5.11956760560145655993787579706, 6.64022863463304124975520830917, 7.40412795099931233113227631024, 8.311462118851943552604800229371, 8.687237935860506230685068708677, 10.01505262797672601510902227339, 10.32185501714346350766328219528
