L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−1.98 − 1.03i)5-s + 1.00·6-s + (0.124 − 0.124i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.666 + 2.13i)10-s + 0.552i·11-s + (−0.707 − 0.707i)12-s + (2.68 − 2.68i)13-s − 0.176·14-s + (2.13 − 0.666i)15-s − 1.00·16-s + (−3.61 + 3.61i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.885 − 0.464i)5-s + 0.408·6-s + (0.0471 − 0.0471i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.210 + 0.674i)10-s + 0.166i·11-s + (−0.204 − 0.204i)12-s + (0.745 − 0.745i)13-s − 0.0471·14-s + (0.551 − 0.172i)15-s − 0.250·16-s + (−0.877 + 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0555 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0555 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.291006 + 0.307654i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.291006 + 0.307654i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.98 + 1.03i)T \) |
| 23 | \( 1 + (4.79 + 0.0741i)T \) |
good | 7 | \( 1 + (-0.124 + 0.124i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.552iT - 11T^{2} \) |
| 13 | \( 1 + (-2.68 + 2.68i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.61 - 3.61i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 29 | \( 1 - 8.68iT - 29T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 37 | \( 1 + (7.79 - 7.79i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + (-1.06 - 1.06i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.597 + 0.597i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.95 + 5.95i)T + 53iT^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 - 9.12iT - 61T^{2} \) |
| 67 | \( 1 + (5.38 - 5.38i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.16T + 71T^{2} \) |
| 73 | \( 1 + (-3.12 + 3.12i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 + (0.190 + 0.190i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.76T + 89T^{2} \) |
| 97 | \( 1 + (1.62 - 1.62i)T - 97iT^{2} \) |
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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
−10.68709018751540786433529004223, −10.05100234663981693234077609674, −8.804256228735691819037760688467, −8.416197064230271949356271479409, −7.38685248154273854652890641164, −6.28156331646722623243529131636, −5.04730558613024352424517448934, −4.09044217058424327509285701382, −3.21800465581829452965402435870, −1.36247656004575622412893771427,
0.29756104258344425686338740685, 2.13216727095801241355923643588, 3.75761444312748762110825987467, 4.83388321801601781176305515176, 6.12266353374578493231448523773, 6.74986499670013118284598707716, 7.60584052634188786734892111925, 8.337791895635327398154257903904, 9.243655705836974586440657210490, 10.28801771490585476249904407745