L(s) = 1 | + (13.8 − 13.8i)2-s + (−128. + 55.7i)3-s + 127. i·4-s + (919. − 1.05e3i)5-s + (−1.01e3 + 2.55e3i)6-s + (6.79e3 + 6.79e3i)7-s + (8.86e3 + 8.86e3i)8-s + (1.34e4 − 1.43e4i)9-s + (−1.83e3 − 2.73e4i)10-s + 2.84e4i·11-s + (−7.07e3 − 1.63e4i)12-s + (2.82e4 − 2.82e4i)13-s + 1.88e5·14-s + (−5.98e4 + 1.86e5i)15-s + 1.80e5·16-s + (−2.11e4 + 2.11e4i)17-s + ⋯ |
L(s) = 1 | + (0.613 − 0.613i)2-s + (−0.917 + 0.397i)3-s + 0.248i·4-s + (0.658 − 0.752i)5-s + (−0.319 + 0.806i)6-s + (1.07 + 1.07i)7-s + (0.765 + 0.765i)8-s + (0.684 − 0.728i)9-s + (−0.0579 − 0.865i)10-s + 0.584i·11-s + (−0.0985 − 0.227i)12-s + (0.273 − 0.273i)13-s + 1.31·14-s + (−0.305 + 0.952i)15-s + 0.690·16-s + (−0.0612 + 0.0612i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.10223 + 0.257165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10223 + 0.257165i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (128. - 55.7i)T \) |
| 5 | \( 1 + (-919. + 1.05e3i)T \) |
good | 2 | \( 1 + (-13.8 + 13.8i)T - 512iT^{2} \) |
| 7 | \( 1 + (-6.79e3 - 6.79e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 - 2.84e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-2.82e4 + 2.82e4i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (2.11e4 - 2.11e4i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 7.59e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (1.18e6 + 1.18e6i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 - 6.93e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (8.18e6 + 8.18e6i)T + 1.29e14iT^{2} \) |
| 41 | \( 1 + 7.44e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (-6.21e6 + 6.21e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-1.95e7 + 1.95e7i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (4.55e7 + 4.55e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 - 3.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.71e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + (1.30e8 + 1.30e8i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 - 1.62e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (1.40e8 - 1.40e8i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 - 4.65e7iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (4.75e8 + 4.75e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 1.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-9.83e8 - 9.83e8i)T + 7.60e17iT^{2} \) |
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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
−17.33167880624206195314410007155, −16.01821382232356246464114941272, −14.31144400575479433663968118474, −12.50257744676345532646495101934, −11.97602887861920986206907422532, −10.38243605927772581841800148433, −8.448660553751480692692102402552, −5.63645994820426355632499007844, −4.48979512228000860729620500285, −1.83567661626376736058409700254,
1.24256083628811256113388763175, 4.71004084941154078260094732568, 6.18779012550941920564825865321, 7.34057316718461635453748729917, 10.33986364248907556730168931469, 11.27550770119864913061143294639, 13.54729446141712830285842287757, 14.08214788647122562189204404101, 15.73149723031189242402157330665, 17.18350115970084991264546519350