| L(s) = 1 | − 1.73·3-s + (3.27 + 3.77i)5-s + 9.55·7-s + 2.99·9-s − 9.92i·11-s − 7.55i·13-s + (−5.67 − 6.54i)15-s − 17.1i·17-s − 26.1i·19-s − 16.5·21-s − 1.67·23-s + (−3.54 + 24.7i)25-s − 5.19·27-s + 0.350·29-s − 46.0i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.654 + 0.755i)5-s + 1.36·7-s + 0.333·9-s − 0.902i·11-s − 0.581i·13-s + (−0.378 − 0.436i)15-s − 1.01i·17-s − 1.37i·19-s − 0.788·21-s − 0.0728·23-s + (−0.141 + 0.989i)25-s − 0.192·27-s + 0.0120·29-s − 1.48i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.923273403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923273403\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-3.27 - 3.77i)T \) |
| good | 7 | \( 1 - 9.55T + 49T^{2} \) |
| 11 | \( 1 + 9.92iT - 121T^{2} \) |
| 13 | \( 1 + 7.55iT - 169T^{2} \) |
| 17 | \( 1 + 17.1iT - 289T^{2} \) |
| 19 | \( 1 + 26.1iT - 361T^{2} \) |
| 23 | \( 1 + 1.67T + 529T^{2} \) |
| 29 | \( 1 - 0.350T + 841T^{2} \) |
| 31 | \( 1 + 46.0iT - 961T^{2} \) |
| 37 | \( 1 - 22.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 77.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 14.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 22.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 94.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 38T + 3.72e3T^{2} \) |
| 67 | \( 1 - 29.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 7.19iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 34.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 46.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 24.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 100.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 131. iT - 9.40e3T^{2} \) |
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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
−9.847289118316707814069815525530, −8.924789686739126160398547243338, −7.938347339568866725601821671586, −7.14906324375420357135941818220, −6.20919685866875512546060064624, −5.35969403340111526714740105203, −4.67731331189571358252967882007, −3.19203358740664302770610764924, −2.10787977866707114990332604731, −0.69440435360764190619165303296,
1.41769419244204653472630830900, 1.91857364169984942664522626921, 3.99263539833658227662398411719, 4.79614838501718739399763771564, 5.47754845116569307519626027765, 6.38798381123117502840199976524, 7.47588319404139339660298683300, 8.337102959159501110710345660646, 9.042632899351982163111528729782, 10.17043833701034043587380869553
