L(s) = 1 | − 1.61·2-s + 1.61·4-s + 7-s − 8-s + 9-s − 1.61·14-s − 1.61·18-s − 1.61·23-s + 25-s + 1.61·28-s + 0.618·29-s + 32-s + 1.61·36-s − 1.61·37-s + 0.618·43-s + 2.61·46-s + 49-s − 1.61·50-s + 0.618·53-s − 56-s − 1.00·58-s + 63-s − 1.61·64-s + 0.618·67-s + 0.618·71-s − 72-s + 2.61·74-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 1.61·4-s + 7-s − 8-s + 9-s − 1.61·14-s − 1.61·18-s − 1.61·23-s + 25-s + 1.61·28-s + 0.618·29-s + 32-s + 1.61·36-s − 1.61·37-s + 0.618·43-s + 2.61·46-s + 49-s − 1.61·50-s + 0.618·53-s − 56-s − 1.00·58-s + 63-s − 1.61·64-s + 0.618·67-s + 0.618·71-s − 72-s + 2.61·74-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5541335636\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5541335636\) |
\(L(1)\) |
|
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 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 - 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 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
−10.36300542543562407087495051759, −9.542526919645445145299868730228, −8.656433858379158665847379669118, −8.029645922169226591581852806740, −7.29147343091261307844789775485, −6.50769438578729732944439167979, −5.11287903436324213387937648395, −4.04389177856401974827877511335, −2.26214367040914603615949696135, −1.28378326549801024809477513366,
1.28378326549801024809477513366, 2.26214367040914603615949696135, 4.04389177856401974827877511335, 5.11287903436324213387937648395, 6.50769438578729732944439167979, 7.29147343091261307844789775485, 8.029645922169226591581852806740, 8.656433858379158665847379669118, 9.542526919645445145299868730228, 10.36300542543562407087495051759