L(s) = 1 | − 8.04·5-s − 2·7-s + 21.7·11-s − 17.3i·13-s − 11.8i·17-s + 10.7i·19-s + 35.0i·23-s + 39.7·25-s − 11.7·29-s − 35.5·31-s + 16.0·35-s + 26i·37-s − 2.28i·41-s + 65.5i·43-s − 27.2i·47-s + ⋯ |
L(s) = 1 | − 1.60·5-s − 0.285·7-s + 1.97·11-s − 1.33i·13-s − 0.697i·17-s + 0.566i·19-s + 1.52i·23-s + 1.59·25-s − 0.405·29-s − 1.14·31-s + 0.459·35-s + 0.702i·37-s − 0.0558i·41-s + 1.52i·43-s − 0.578i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.103430954\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.103430954\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 8.04T + 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 - 21.7T + 121T^{2} \) |
| 13 | \( 1 + 17.3iT - 169T^{2} \) |
| 17 | \( 1 + 11.8iT - 289T^{2} \) |
| 19 | \( 1 - 10.7iT - 361T^{2} \) |
| 23 | \( 1 - 35.0iT - 529T^{2} \) |
| 29 | \( 1 + 11.7T + 841T^{2} \) |
| 31 | \( 1 + 35.5T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 2.28iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 65.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 27.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 7.52iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 84.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 84.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 75.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 48.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 106.T + 9.40e3T^{2} \) |
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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
−8.610851064473149004552237810344, −7.75436029788258549600554386546, −7.30902533362595070708023909835, −6.43328654157516262313846336707, −5.48526243138319226888683391594, −4.45516104593447947688293822688, −3.55284657131560844711127827345, −3.29902685830682125157628359516, −1.46676758524875995021299727077, −0.37409109466220892079051889817,
0.871844282124750169091813733854, 2.16144697075418172191209092738, 3.77299648221278305578233263906, 3.86075618318158765294155011738, 4.70204033195320994338783982850, 6.09562135819496767114467594573, 6.92946281281943833360004508933, 7.17919933914281772323565951675, 8.453455497144231780537497328317, 8.833771032023044703486605146348