L(s) = 1 | + (−0.591 + 3.95i)2-s − 4.43i·3-s + (−15.3 − 4.67i)4-s + 13.1·5-s + (17.5 + 2.62i)6-s + 8.96i·7-s + (27.5 − 57.7i)8-s + 61.3·9-s + (−7.74 + 51.8i)10-s + 193. i·11-s + (−20.7 + 67.8i)12-s + 173.·13-s + (−35.4 − 5.30i)14-s − 58.1i·15-s + (212. + 143. i)16-s + 83.2·17-s + ⋯ |
L(s) = 1 | + (−0.147 + 0.989i)2-s − 0.492i·3-s + (−0.956 − 0.292i)4-s + 0.524·5-s + (0.487 + 0.0728i)6-s + 0.183i·7-s + (0.430 − 0.902i)8-s + 0.757·9-s + (−0.0774 + 0.518i)10-s + 1.60i·11-s + (−0.144 + 0.471i)12-s + 1.02·13-s + (−0.181 − 0.0270i)14-s − 0.258i·15-s + (0.828 + 0.559i)16-s + 0.288·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.29682 + 0.959558i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29682 + 0.959558i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.591 - 3.95i)T \) |
| 19 | \( 1 - 82.8iT \) |
good | 3 | \( 1 + 4.43iT - 81T^{2} \) |
| 5 | \( 1 - 13.1T + 625T^{2} \) |
| 7 | \( 1 - 8.96iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 193. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 83.2T + 8.35e4T^{2} \) |
| 23 | \( 1 - 684. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.43e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 750. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 437.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 895.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 928. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.11e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.58e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 306. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.09e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.56e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 149. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.82e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.89e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.84e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.71e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 2.17e3T + 8.85e7T^{2} \) |
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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
−13.94100134599979158779436808976, −13.17424356716419836437807869603, −12.16195439757423528827997029917, −10.20481861534473774842825932292, −9.421115254988940967921317700524, −7.939031474868509598259479812353, −6.96231067008053316061936974961, −5.81953595600877153971760083022, −4.30275657478037660597959461766, −1.52537212064785729892270648952,
1.08061512364589354540739467730, 3.13001905177984786114481320750, 4.48811336655989142007723214343, 6.10686239702446974298796468670, 8.216064385980317278544916231083, 9.225159280277222179505447717865, 10.41488792683484374501328454035, 11.00998382690325840312140581254, 12.41357853957390639087610809925, 13.53836684010208327426760024926