| 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
−10.32185501714346350766328219528, −10.01505262797672601510902227339, −8.687237935860506230685068708677, −8.311462118851943552604800229371, −7.40412795099931233113227631024, −6.64022863463304124975520830917, −5.11956760560145655993787579706, −4.15163065452570825223617159750, −3.05951936153958769031781583977, −1.51196139799478377140498685523,
0.53497467625634369192266224158, 2.43355868710672843142917510633, 3.35016104939503702275097425335, 4.45134272387735040144317043397, 6.19999338194197618340334965365, 6.77724071066080495114365132431, 7.926443635069747378796580534798, 8.518788909739312906338660095962, 9.055499804833808007754715725031, 10.31949210347229145346714224988
