| L(s) = 1 | + i·3-s − 7-s − i·11-s + (−1 + i)17-s + (−1 + i)19-s − i·21-s + (−1 + i)23-s + i·25-s + i·27-s + (1 + i)29-s + 33-s + i·37-s − i·41-s + 47-s + (−1 − i)51-s + ⋯ |
| L(s) = 1 | + i·3-s − 7-s − i·11-s + (−1 + i)17-s + (−1 + i)19-s − i·21-s + (−1 + i)23-s + i·25-s + i·27-s + (1 + i)29-s + 33-s + i·37-s − i·41-s + 47-s + (−1 − i)51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7568806344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7568806344\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (1 - i)T - iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (1 + i)T + iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.507444227413182639724242237583, −8.793859658400093389506068012217, −8.179978204166365741812017994939, −7.02315924906918231624602191194, −6.22626700897869833321045673292, −5.63132072761486984052024572822, −4.51788677214061706336735399163, −3.72609844272081899159515164610, −3.22779171878161646874349107106, −1.73348169387667460872441360762,
0.48634662600052422695110422645, 2.20957810832280323988694214502, 2.59478174855576320291414965767, 4.20866552620871849574870979456, 4.67938280251218641218901462148, 6.23181389123048806553640298027, 6.53859801680196118896579963242, 7.15972804931450572252942872587, 7.996721827830266799508879916376, 8.807278911850031532327764001580
