| L(s) = 1 | + (0.203 − 1.39i)2-s + (0.487 + 0.957i)3-s + (−1.91 − 0.569i)4-s + (0.870 + 2.05i)5-s + (1.43 − 0.487i)6-s + (−0.707 − 0.707i)7-s + (−1.18 + 2.56i)8-s + (1.08 − 1.49i)9-s + (3.05 − 0.798i)10-s + (2.48 + 3.42i)11-s + (−0.389 − 2.11i)12-s + (−0.145 − 0.920i)13-s + (−1.13 + 0.845i)14-s + (−1.54 + 1.83i)15-s + (3.35 + 2.18i)16-s + (−3.77 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.143 − 0.989i)2-s + (0.281 + 0.552i)3-s + (−0.958 − 0.284i)4-s + (0.389 + 0.921i)5-s + (0.587 − 0.199i)6-s + (−0.267 − 0.267i)7-s + (−0.419 + 0.907i)8-s + (0.361 − 0.497i)9-s + (0.967 − 0.252i)10-s + (0.749 + 1.03i)11-s + (−0.112 − 0.610i)12-s + (−0.0404 − 0.255i)13-s + (−0.302 + 0.226i)14-s + (−0.399 + 0.474i)15-s + (0.837 + 0.546i)16-s + (−0.914 − 0.466i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69008 + 0.139847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69008 + 0.139847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.203 + 1.39i)T \) |
| 5 | \( 1 + (-0.870 - 2.05i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (-0.487 - 0.957i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-2.48 - 3.42i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.145 + 0.920i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.77 + 1.92i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.08 - 6.41i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.02 - 6.46i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-3.50 - 1.13i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.87 + 2.23i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-11.6 + 1.83i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (4.81 + 3.49i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.823 - 0.823i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.00 - 2.55i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.759 + 0.387i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (6.96 + 5.05i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (6.65 - 4.83i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.28 + 6.43i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 3.59i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.54 - 0.402i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.27 - 3.90i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (13.5 + 6.88i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-1.03 - 1.42i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.13 + 6.15i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.26358736409473546620819208897, −9.689043949641683000350987579069, −9.422203876324825565224328687364, −8.000492863758505675778974260419, −6.87500747358963932273729855929, −5.94391194662295531795508875182, −4.56859504983247466410089163056, −3.78541919570341379462894282106, −2.90283058488405969252914358949, −1.57528260784210807330558902488,
0.933866297556365593363799573146, 2.68291101856787715354939726010, 4.33689711767371561209228028932, 4.95548447183052448914664948733, 6.35259602632671086335090594949, 6.57713208988770997137408695504, 7.978732733690205391267833007208, 8.574087460650817867948087706077, 9.152940382721035112572021050825, 10.11635835579585158957502398442
