| L(s) = 1 | − 0.618·2-s − 0.618·4-s − 7-s + 8-s + 9-s + 0.618·14-s − 0.618·18-s + 0.618·23-s + 25-s + 0.618·28-s + 1.61·29-s − 0.999·32-s − 0.618·36-s + 0.618·37-s + 1.61·43-s − 0.381·46-s + 49-s − 0.618·50-s − 1.61·53-s − 56-s − 1.00·58-s − 63-s + 0.618·64-s − 1.61·67-s − 1.61·71-s + 72-s − 0.381·74-s + ⋯ |
| L(s) = 1 | − 0.618·2-s − 0.618·4-s − 7-s + 8-s + 9-s + 0.618·14-s − 0.618·18-s + 0.618·23-s + 25-s + 0.618·28-s + 1.61·29-s − 0.999·32-s − 0.618·36-s + 0.618·37-s + 1.61·43-s − 0.381·46-s + 49-s − 0.618·50-s − 1.61·53-s − 56-s − 1.00·58-s − 63-s + 0.618·64-s − 1.61·67-s − 1.61·71-s + 72-s − 0.381·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6164242913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6164242913\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 - 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.29435991682558834056466196754, −9.447722241694895253734364049489, −8.970635093423396749631420443901, −7.906725143349619261563022862851, −7.09002114673892051579501480244, −6.24523904327756217271547276848, −4.90942300525962899298205958434, −4.13921915940365722561338672796, −2.90489851578749753955373186149, −1.11951979201372306401125473864,
1.11951979201372306401125473864, 2.90489851578749753955373186149, 4.13921915940365722561338672796, 4.90942300525962899298205958434, 6.24523904327756217271547276848, 7.09002114673892051579501480244, 7.906725143349619261563022862851, 8.970635093423396749631420443901, 9.447722241694895253734364049489, 10.29435991682558834056466196754
