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 + (−27.7 + 51.0i)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 + (−0.478 + 0.878i)15-s − 0.492·16-s − 0.437i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.100028 - 0.880467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100028 - 0.880467i\) |
\(L(\frac{5}{2})\) |
|
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
−12.82405197606328126973800002231, −11.91569087508402684027820621049, −11.41195413259346570757475179046, −9.153720561649930114279106820727, −8.356757689920755744882993414956, −7.11670247895136590604202024557, −5.46129717935941805547611066404, −4.75935366893241544348865199861, −2.76696581141766306686932678939, −0.39134055827673701660836069884,
3.27567668605156166817625915330, 4.37468744296815292841652931447, 5.16137327129171854848695849492, 6.94486854714627642318027873551, 8.285373012354724123008427414310, 9.751233363834472891859132160978, 10.62518432552142824853159497609, 11.76006138756578346048807035376, 12.60424869671316495199001642711, 14.19612769828899848990719318872