| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s + 11-s + 2·13-s − 4·14-s + 16-s + 4·19-s + 20-s + 22-s − 8·23-s + 25-s + 2·26-s − 4·28-s + 2·29-s + 32-s − 4·35-s + 2·37-s + 4·38-s + 40-s + 6·41-s + 4·43-s + 44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.917·19-s + 0.223·20-s + 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 0.371·29-s + 0.176·32-s − 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.150·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.93196059202256, −12.62941073708695, −12.07190447757386, −11.79700954631918, −11.16173480677109, −10.70707575159451, −10.06766821041050, −9.787804526161555, −9.519488203913571, −8.733858082349250, −8.472889046287083, −7.552524906694871, −7.359704367028488, −6.664916956551768, −6.236049349594578, −5.881778565374540, −5.613865862482173, −4.789864103891409, −4.236956697145191, −3.709842136391797, −3.363547778556554, −2.689846704359564, −2.320738719619616, −1.482194328027519, −0.8696901327300646, 0,
0.8696901327300646, 1.482194328027519, 2.320738719619616, 2.689846704359564, 3.363547778556554, 3.709842136391797, 4.236956697145191, 4.789864103891409, 5.613865862482173, 5.881778565374540, 6.236049349594578, 6.664916956551768, 7.359704367028488, 7.552524906694871, 8.472889046287083, 8.733858082349250, 9.519488203913571, 9.787804526161555, 10.06766821041050, 10.70707575159451, 11.16173480677109, 11.79700954631918, 12.07190447757386, 12.62941073708695, 12.93196059202256
