| L(s) = 1 | − 2·5-s + 4·11-s − 2·17-s + 4·19-s + 8·23-s − 25-s − 6·29-s − 8·31-s − 6·37-s − 6·41-s + 4·43-s − 7·49-s + 2·53-s − 8·55-s + 4·59-s − 2·61-s + 4·67-s + 8·71-s − 10·73-s − 8·79-s − 4·83-s + 4·85-s − 6·89-s − 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s − 1.07·55-s + 0.520·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 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
−16.83562863883016, −15.92698678047981, −15.57906158754720, −14.92953713531072, −14.46926674229150, −13.91944536530786, −13.06052437607871, −12.75411225879171, −11.88885978552740, −11.46322570078511, −11.15536185490757, −10.34771752586776, −9.503690354710551, −9.047079994351056, −8.568396788237467, −7.665589421442135, −7.155380972173833, −6.743276567479184, −5.772484236425666, −5.128175078489808, −4.360804668851595, −3.601721758503849, −3.260187228168582, −1.994910160164052, −1.153278006863288, 0,
1.153278006863288, 1.994910160164052, 3.260187228168582, 3.601721758503849, 4.360804668851595, 5.128175078489808, 5.772484236425666, 6.743276567479184, 7.155380972173833, 7.665589421442135, 8.568396788237467, 9.047079994351056, 9.503690354710551, 10.34771752586776, 11.15536185490757, 11.46322570078511, 11.88885978552740, 12.75411225879171, 13.06052437607871, 13.91944536530786, 14.46926674229150, 14.92953713531072, 15.57906158754720, 15.92698678047981, 16.83562863883016
