| L(s) = 1 | + (−0.719 + 1.21i)2-s + (−1.27 − 2.50i)3-s + (−0.964 − 1.75i)4-s + (−1.94 + 1.09i)5-s + (3.96 + 0.248i)6-s + (0.707 + 0.707i)7-s + (2.82 + 0.0871i)8-s + (−2.88 + 3.97i)9-s + (0.0628 − 3.16i)10-s + (0.113 + 0.156i)11-s + (−3.16 + 4.65i)12-s + (0.311 + 1.96i)13-s + (−1.36 + 0.351i)14-s + (5.24 + 3.47i)15-s + (−2.14 + 3.37i)16-s + (2.57 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.508 + 0.860i)2-s + (−0.737 − 1.44i)3-s + (−0.482 − 0.876i)4-s + (−0.870 + 0.491i)5-s + (1.62 + 0.101i)6-s + (0.267 + 0.267i)7-s + (0.999 + 0.0308i)8-s + (−0.962 + 1.32i)9-s + (0.0198 − 0.999i)10-s + (0.0342 + 0.0471i)11-s + (−0.912 + 1.34i)12-s + (0.0864 + 0.545i)13-s + (−0.366 + 0.0940i)14-s + (1.35 + 0.897i)15-s + (−0.535 + 0.844i)16-s + (0.625 + 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.653103 - 0.127414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.653103 - 0.127414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.719 - 1.21i)T \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| good | 3 | \( 1 + (1.27 + 2.50i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-0.113 - 0.156i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.311 - 1.96i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.57 - 1.31i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 3.36i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.18 + 7.46i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (6.16 + 2.00i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.24 + 0.405i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.90 + 1.09i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.540 - 0.392i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.00 + 8.00i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.53 + 1.29i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (2.57 - 1.31i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (3.06 + 2.22i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.48 + 2.53i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.15 + 4.22i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-4.27 - 1.39i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 2.15i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.57 + 7.92i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.53 - 1.80i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 14.9i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.65 + 13.0i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.60484840216684377294108588899, −9.308613128166668029652578695737, −8.116025063315368127348913750457, −7.78708867567341840707833668435, −6.88911733001018680420917822194, −6.27272477698881825593960105523, −5.41379442701519066889123147628, −4.10912201659125973455522685523, −2.12746253203311123016055088670, −0.69213932189769090086956674231,
0.879282095600709747999777533000, 3.16583496561545867511977167227, 3.94673271573944688471038010937, 4.78528414082470934930265373851, 5.53856245518686621204701984181, 7.36232374534202152634750960943, 8.083473848899809373999542532424, 9.324179747618769032977649871029, 9.538229369620891017159353914961, 10.70830520752634262060169668522
