L(s) = 1 | + (−1.5 − 2.59i)3-s + (4.91 − 8.50i)5-s + (−4.5 + 7.79i)9-s + (7.08 + 12.2i)11-s + 26.1·13-s − 29.4·15-s + (−39.2 − 68.0i)17-s + (36.5 − 63.3i)19-s + (48 − 83.1i)23-s + (14.2 + 24.6i)25-s + 27·27-s + 173.·29-s + (33.6 + 58.2i)31-s + (21.2 − 36.8i)33-s + (150. − 261. i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.439 − 0.761i)5-s + (−0.166 + 0.288i)9-s + (0.194 + 0.336i)11-s + 0.557·13-s − 0.507·15-s + (−0.560 − 0.971i)17-s + (0.441 − 0.765i)19-s + (0.435 − 0.753i)23-s + (0.113 + 0.197i)25-s + 0.192·27-s + 1.10·29-s + (0.194 + 0.337i)31-s + (0.112 − 0.194i)33-s + (0.670 − 1.16i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.610531701\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.610531701\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-4.91 + 8.50i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7.08 - 12.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 26.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + (39.2 + 68.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-36.5 + 63.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-48 + 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-33.6 - 58.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (45.5 - 78.9i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-81.6 - 141. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (300. + 520. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-285. + 495. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-269. - 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (221. + 383. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-22.8 + 39.5i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 686.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-330. + 571. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 658.T + 9.12e5T^{2} \) |
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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
−9.898688802721141716381804183332, −9.020924290856002655828733276121, −8.360367244741489754631686781695, −7.10719031894088817988546950690, −6.46607861742881526716299339979, −5.24234120158685951583743009332, −4.62057047159707113028859450741, −2.97860823145643375278612658508, −1.64741665556912293653378585974, −0.51518012410538863732201885457,
1.41679011685305680249997197603, 2.94650425930451893801797998430, 3.88070649834038355842515099179, 5.11280721387812438938278068994, 6.18697242892464661326049025264, 6.69123885743128648219891912420, 8.088349794807743316634467267022, 8.864299276462659707209233146907, 10.08997644722540324480265069236, 10.36080266653139261017259074555