| L(s) = 1 | + (−1.36 + 0.380i)2-s + (−0.743 − 1.45i)3-s + (1.71 − 1.03i)4-s + (2.23 − 0.136i)5-s + (1.56 + 1.70i)6-s + (−0.707 − 0.707i)7-s + (−1.93 + 2.06i)8-s + (0.186 − 0.256i)9-s + (−2.98 + 1.03i)10-s + (3.05 + 4.20i)11-s + (−2.78 − 1.72i)12-s + (0.994 + 6.27i)13-s + (1.23 + 0.694i)14-s + (−1.85 − 3.15i)15-s + (1.85 − 3.54i)16-s + (−0.476 − 0.242i)17-s + ⋯ |
| L(s) = 1 | + (−0.963 + 0.268i)2-s + (−0.429 − 0.842i)3-s + (0.855 − 0.518i)4-s + (0.998 − 0.0608i)5-s + (0.640 + 0.696i)6-s + (−0.267 − 0.267i)7-s + (−0.684 + 0.729i)8-s + (0.0620 − 0.0854i)9-s + (−0.944 + 0.327i)10-s + (0.921 + 1.26i)11-s + (−0.803 − 0.498i)12-s + (0.275 + 1.74i)13-s + (0.329 + 0.185i)14-s + (−0.479 − 0.814i)15-s + (0.463 − 0.886i)16-s + (−0.115 − 0.0589i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02323 + 0.165590i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02323 + 0.165590i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 - 0.380i)T \) |
| 5 | \( 1 + (-2.23 + 0.136i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (0.743 + 1.45i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-3.05 - 4.20i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.994 - 6.27i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (0.476 + 0.242i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.55 - 7.85i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.394 - 2.48i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.69 + 0.874i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.33 - 0.434i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.17 + 0.186i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (3.19 + 2.32i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.04 + 1.04i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.06 + 3.59i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (12.0 - 6.14i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-6.47 - 4.70i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 7.51i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.728 + 1.42i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (8.29 + 2.69i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-11.2 - 1.77i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.877 + 2.70i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.25 - 1.65i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (0.0577 + 0.0794i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.06 + 9.93i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.14896760698702398825458317625, −9.556963925261882983669111404081, −9.053595338775805149358950516787, −7.65967917108466506199457823438, −6.87429458767960716656182717057, −6.44554790752553893044308203002, −5.56114390457557455469175282154, −3.97473172459740940667178233018, −1.88799607567671531459328446843, −1.47614043257819756260295311080,
0.866898681806173019374027816340, 2.61140629324466900678412406802, 3.55127425812109610040634188200, 5.17961751286527068880496902123, 5.95293355059195223208204705162, 6.82098919774327229704162100843, 8.095933591287993727795860685176, 9.024821904478911622788172905668, 9.538666849989632799258281152560, 10.41762074588773906927586938135
