| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s − 4·11-s − 14-s + 16-s − 6·17-s + 4·19-s + 2·20-s − 4·22-s − 25-s − 28-s + 6·29-s + 8·31-s + 32-s − 6·34-s − 2·35-s − 10·37-s + 4·38-s + 2·40-s − 6·41-s + 4·43-s − 4·44-s + 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 1.64·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.82510666025973, −15.52906989347555, −14.70615238608207, −13.88527044324473, −13.63223057265161, −13.44565933663405, −12.58021843800449, −12.25089578510138, −11.53091204947293, −10.83825183646276, −10.40197281523711, −9.854002847050240, −9.325959722145717, −8.487880356461537, −8.029772206832952, −7.141933866893812, −6.639823731110567, −6.130490238153163, −5.373900508270412, −4.992744357999828, −4.299204637597500, −3.385539714159941, −2.679086541484318, −2.247475309757218, −1.284653753647193, 0,
1.284653753647193, 2.247475309757218, 2.679086541484318, 3.385539714159941, 4.299204637597500, 4.992744357999828, 5.373900508270412, 6.130490238153163, 6.639823731110567, 7.141933866893812, 8.029772206832952, 8.487880356461537, 9.325959722145717, 9.854002847050240, 10.40197281523711, 10.83825183646276, 11.53091204947293, 12.25089578510138, 12.58021843800449, 13.44565933663405, 13.63223057265161, 13.88527044324473, 14.70615238608207, 15.52906989347555, 15.82510666025973
