L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.28 + 1.83i)5-s + (2.64 − 0.0564i)7-s + (−0.707 − 0.707i)8-s + (0.386 + 2.20i)10-s + 1.41·11-s + (0.772 − 0.772i)13-s + (1.83 − 1.91i)14-s − 1.00·16-s + (0.546 + 0.546i)17-s + 5.95·19-s + (1.83 + 1.28i)20-s + (1.00 − 1.00i)22-s + (5.23 + 5.23i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.574 + 0.818i)5-s + (0.999 − 0.0213i)7-s + (−0.250 − 0.250i)8-s + (0.122 + 0.696i)10-s + 0.426·11-s + (0.214 − 0.214i)13-s + (0.489 − 0.510i)14-s − 0.250·16-s + (0.132 + 0.132i)17-s + 1.36·19-s + (0.409 + 0.287i)20-s + (0.213 − 0.213i)22-s + (1.09 + 1.09i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99561 - 0.361725i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99561 - 0.361725i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1.28 - 1.83i)T \) |
| 7 | \( 1 + (-2.64 + 0.0564i)T \) |
good | 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + (-0.772 + 0.772i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.546 - 0.546i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 + (-5.23 - 5.23i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.992iT - 29T^{2} \) |
| 31 | \( 1 - 2.85iT - 31T^{2} \) |
| 37 | \( 1 + (-5.70 + 5.70i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.18iT - 41T^{2} \) |
| 43 | \( 1 + (2 + 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.32 + 7.32i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.40 - 2.40i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 3.63iT - 61T^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.06T + 71T^{2} \) |
| 73 | \( 1 + (11.3 - 11.3i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.40iT - 79T^{2} \) |
| 83 | \( 1 + (-8.79 + 8.79i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + (7.76 + 7.76i)T + 97iT^{2} \) |
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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
−10.80961178130202954102430514670, −9.908955547675772361120401230333, −8.868274657883202562642647052733, −7.74032591237099545092717642377, −7.09302211648210025724983542578, −5.84584405892629011275038075782, −4.89689458761293683978336367488, −3.79607174852778426657431048307, −2.91050164366370608163297888495, −1.37715834203586037201825955911,
1.27223074576835704841183885195, 3.13412137372153330804087718546, 4.44568604114655367238924660506, 4.91486846516340515630576911068, 6.01371033816151782336211033613, 7.22460762085316463334169740383, 7.939834202282919344947400129848, 8.719074663561429556400604206399, 9.510779447052571332526678837042, 10.95528828144386880023362179558