| L(s) = 1 | + 1.17·2-s + 0.381·4-s + 0.618·7-s − 0.726·8-s + 1.61·13-s + 0.726·14-s − 1.23·16-s + 1.90·17-s − 1.61·19-s − 1.17·23-s + 25-s + 1.90·26-s + 0.236·28-s − 1.90·29-s − 0.726·32-s + 2.23·34-s − 1.90·38-s − 1.17·41-s − 1.38·46-s − 0.618·49-s + 1.17·50-s + 0.618·52-s − 0.449·56-s − 2.23·58-s − 1.90·59-s − 0.618·61-s + 0.381·64-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.381·4-s + 0.618·7-s − 0.726·8-s + 1.61·13-s + 0.726·14-s − 1.23·16-s + 1.90·17-s − 1.61·19-s − 1.17·23-s + 25-s + 1.90·26-s + 0.236·28-s − 1.90·29-s − 0.726·32-s + 2.23·34-s − 1.90·38-s − 1.17·41-s − 1.38·46-s − 0.618·49-s + 1.17·50-s + 0.618·52-s − 0.449·56-s − 2.23·58-s − 1.90·59-s − 0.618·61-s + 0.381·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.749781979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749781979\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 - 1.90T + T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 + 1.17T + T^{2} \) |
| 29 | \( 1 + 1.90T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.17T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.90T + T^{2} \) |
| 61 | \( 1 + 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 - 1.90T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−10.56269559610793602224117442993, −9.387892297994147839452815589152, −8.500498904261277678789338432181, −7.80837220789313540091478005008, −6.43681024640810104622541348410, −5.84419107674818716313263069884, −4.98298982328206976453977122362, −3.95424233861339061011439569191, −3.32449946038957578896279548384, −1.74545169287564529552992587700,
1.74545169287564529552992587700, 3.32449946038957578896279548384, 3.95424233861339061011439569191, 4.98298982328206976453977122362, 5.84419107674818716313263069884, 6.43681024640810104622541348410, 7.80837220789313540091478005008, 8.500498904261277678789338432181, 9.387892297994147839452815589152, 10.56269559610793602224117442993
