| L(s) = 1 | + 2-s − 4-s − 3·8-s − 2·11-s − 16-s − 6·19-s − 2·22-s − 6·23-s + 6·29-s − 6·31-s + 5·32-s − 6·37-s − 6·38-s − 8·41-s − 6·43-s + 2·44-s − 6·46-s − 8·47-s − 7·49-s − 12·53-s + 6·58-s + 2·59-s + 6·61-s − 6·62-s + 7·64-s + 12·67-s − 2·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.603·11-s − 1/4·16-s − 1.37·19-s − 0.426·22-s − 1.25·23-s + 1.11·29-s − 1.07·31-s + 0.883·32-s − 0.986·37-s − 0.973·38-s − 1.24·41-s − 0.914·43-s + 0.301·44-s − 0.884·46-s − 1.16·47-s − 49-s − 1.64·53-s + 0.787·58-s + 0.260·59-s + 0.768·61-s − 0.762·62-s + 7/8·64-s + 1.46·67-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4194434256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4194434256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.63615801417995, −14.33770430755100, −13.83234476572462, −13.24267203602763, −12.76454283942022, −12.50650620846098, −11.79287753592975, −11.31385404049718, −10.61408328662590, −9.960361343469975, −9.772627148481799, −8.819852481029797, −8.343769270098931, −8.130065640635924, −7.141675645127153, −6.473435353915248, −6.115705311487228, −5.291255899575442, −4.903509434311098, −4.330087301250740, −3.600007979475042, −3.172594046842404, −2.256507811376516, −1.603431162877458, −0.2030468803097096,
0.2030468803097096, 1.603431162877458, 2.256507811376516, 3.172594046842404, 3.600007979475042, 4.330087301250740, 4.903509434311098, 5.291255899575442, 6.115705311487228, 6.473435353915248, 7.141675645127153, 8.130065640635924, 8.343769270098931, 8.819852481029797, 9.772627148481799, 9.960361343469975, 10.61408328662590, 11.31385404049718, 11.79287753592975, 12.50650620846098, 12.76454283942022, 13.24267203602763, 13.83234476572462, 14.33770430755100, 14.63615801417995
