| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 5·11-s − 2·14-s + 16-s + 2·17-s − 2·19-s − 20-s + 5·22-s + 23-s + 25-s − 2·28-s − 5·29-s − 11·31-s + 32-s + 2·34-s + 2·35-s + 3·37-s − 2·38-s − 40-s + 2·41-s − 11·43-s + 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s + 1.06·22-s + 0.208·23-s + 1/5·25-s − 0.377·28-s − 0.928·29-s − 1.97·31-s + 0.176·32-s + 0.342·34-s + 0.338·35-s + 0.493·37-s − 0.324·38-s − 0.158·40-s + 0.312·41-s − 1.67·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−16.14720448964764, −15.92330499338613, −14.86435841185941, −14.67055261877605, −14.36498495760336, −13.34863054766657, −12.92937072132108, −12.58235806750594, −11.79156017018372, −11.35216667621180, −10.98366146534152, −9.810329638704513, −9.741685270285520, −8.827373453697771, −8.281468017675577, −7.404743055922368, −6.827329483012723, −6.460393379012897, −5.652620734622888, −5.061649942748317, −4.087785359245757, −3.695080155525444, −3.186298700794338, −2.088498202727563, −1.286660100608076, 0,
1.286660100608076, 2.088498202727563, 3.186298700794338, 3.695080155525444, 4.087785359245757, 5.061649942748317, 5.652620734622888, 6.460393379012897, 6.827329483012723, 7.404743055922368, 8.281468017675577, 8.827373453697771, 9.741685270285520, 9.810329638704513, 10.98366146534152, 11.35216667621180, 11.79156017018372, 12.58235806750594, 12.92937072132108, 13.34863054766657, 14.36498495760336, 14.67055261877605, 14.86435841185941, 15.92330499338613, 16.14720448964764
