| L(s) = 1 | − i·3-s + (2 − i)5-s + 2i·7-s − 9-s + 2·11-s − 6i·13-s + (−1 − 2i)15-s − 2i·17-s + 2·21-s + 4i·23-s + (3 − 4i)25-s + i·27-s + 8·31-s − 2i·33-s + (2 + 4i)35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s + 0.755i·7-s − 0.333·9-s + 0.603·11-s − 1.66i·13-s + (−0.258 − 0.516i)15-s − 0.485i·17-s + 0.436·21-s + 0.834i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s + 1.43·31-s − 0.348i·33-s + (0.338 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59631 - 0.986577i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59631 - 0.986577i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.833776562646347466676882747811, −8.971978897568705580423844699306, −8.322158042952626455600158208771, −7.37690217813721676929007350833, −6.26610215086726361520580870028, −5.66263154031096757822641733587, −4.87625642194862086874035110455, −3.24016238096554676569682538509, −2.25673511107964657709754938345, −0.975660376270329757854206180532,
1.50590960483716422198531697519, 2.79037509052512667748577503659, 4.08235065221894392875316020330, 4.67011774542556332856712106536, 6.12787929568860191645301468827, 6.54196454830521372494982997938, 7.55950415037675494715112703776, 8.826894411114671847087748345650, 9.384064569249551478990180713425, 10.18364502159992335238648205135
