| L(s) = 1 | + 3-s − 2.44i·5-s + (−1 − 2.44i)7-s + 9-s + 2.44i·11-s − 4.89i·13-s − 2.44i·15-s − 2.44i·17-s − 2·19-s + (−1 − 2.44i)21-s + 7.34i·23-s − 0.999·25-s + 27-s + 6·29-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.09i·5-s + (−0.377 − 0.925i)7-s + 0.333·9-s + 0.738i·11-s − 1.35i·13-s − 0.632i·15-s − 0.594i·17-s − 0.458·19-s + (−0.218 − 0.534i)21-s + 1.53i·23-s − 0.199·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24288 - 0.835062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24288 - 0.835062i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
| good | 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 11 | \( 1 - 2.44iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7.34iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 9.79iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 12.2iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−11.43679171056899787541640343885, −10.06766355968885829520699300951, −9.717392401845761194423521744001, −8.440458048898924551875581166399, −7.78317009940420519828714069374, −6.69933743306066731178228268093, −5.20535985790215078215421276713, −4.29576391185447612020781873040, −2.98780807657101768177423559533, −1.08573253370453366231816612177,
2.28433892062814274437299330081, 3.19122070782587407155894352746, 4.54391613662404546447331641770, 6.35383654374002150158209640459, 6.59115556682638326977842561465, 8.173470128927599364400138264031, 8.825212501194761584037324141507, 9.880561869990121237884752419881, 10.76496468190376199933363647140, 11.69379779201495792356868184488
