L(s) = 1 | − 2.24·2-s + (−1.06 − 5.08i)3-s − 2.96·4-s + (8.51 − 7.24i)5-s + (2.39 + 11.4i)6-s + (12.2 − 13.8i)7-s + 24.6·8-s + (−24.7 + 10.8i)9-s + (−19.1 + 16.2i)10-s − 25.7i·11-s + (3.16 + 15.0i)12-s − 68.2·13-s + (−27.5 + 31.1i)14-s + (−45.9 − 35.5i)15-s − 31.5·16-s − 30.6i·17-s + ⋯ |
L(s) = 1 | − 0.793·2-s + (−0.205 − 0.978i)3-s − 0.370·4-s + (0.761 − 0.648i)5-s + (0.162 + 0.776i)6-s + (0.662 − 0.748i)7-s + 1.08·8-s + (−0.915 + 0.401i)9-s + (−0.604 + 0.514i)10-s − 0.705i·11-s + (0.0760 + 0.362i)12-s − 1.45·13-s + (−0.526 + 0.594i)14-s + (−0.790 − 0.612i)15-s − 0.492·16-s − 0.437i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0623221 - 0.663211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0623221 - 0.663211i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.06 + 5.08i)T \) |
| 5 | \( 1 + (-8.51 + 7.24i)T \) |
| 7 | \( 1 + (-12.2 + 13.8i)T \) |
good | 2 | \( 1 + 2.24T + 8T^{2} \) |
| 11 | \( 1 + 25.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 68.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 30.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 109. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 16.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 81.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 0.366iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 5.95T + 1.48e5T^{2} \) |
| 59 | \( 1 + 198.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 83.5iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 1.08e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 773. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 448.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 429. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 5.29T + 7.04e5T^{2} \) |
| 97 | \( 1 + 435.T + 9.12e5T^{2} \) |
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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
−12.84079096988667102019423814563, −11.80336964608270127746565552933, −10.42075773109374130610590949518, −9.516065483669436028112928441470, −8.174015299669826950542147744998, −7.62588036238537419661519341880, −5.93777463290084531999619007055, −4.64021306863339162422532007398, −1.86008278719930424667001421788, −0.49656403455112526116404887303,
2.33477200278930937463082362733, 4.52409657781549335088449738671, 5.52928613205178582298024964940, 7.28305663779283533114627638454, 8.731533395056729011240143314080, 9.586164755485076484489492659129, 10.25906577199517939988623305533, 11.27884046194681657844488896294, 12.60372180101717701578917522823, 14.19644649685016480443740486887