L(s) = 1 | − 2.64·2-s + 5.00·4-s + 3·7-s − 7.93·8-s + 5.29·11-s − 2·13-s − 7.93·14-s + 11.0·16-s + 5.29·17-s + 19-s − 14.0·22-s + 5.29·26-s + 15.0·28-s − 5.29·29-s − 3·31-s − 13.2·32-s − 14.0·34-s − 37-s − 2.64·38-s − 5.29·41-s + 43-s + 26.4·44-s + 5.29·47-s + 2·49-s − 10.0·52-s + 5.29·53-s − 23.8·56-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 2.50·4-s + 1.13·7-s − 2.80·8-s + 1.59·11-s − 0.554·13-s − 2.12·14-s + 2.75·16-s + 1.28·17-s + 0.229·19-s − 2.98·22-s + 1.03·26-s + 2.83·28-s − 0.982·29-s − 0.538·31-s − 2.33·32-s − 2.40·34-s − 0.164·37-s − 0.429·38-s − 0.826·41-s + 0.152·43-s + 3.98·44-s + 0.771·47-s + 0.285·49-s − 1.38·52-s + 0.726·53-s − 3.18·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8108172587\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8108172587\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−10.29782179163196724125717137600, −9.476891964606973669092818124945, −8.899608505640944065032256426165, −7.955187968227007460564027584057, −7.38888700808618748687935596601, −6.46416894052052834655430003192, −5.28988083268399132293722298205, −3.64271201299328746371537786528, −2.03451897170989253356148732431, −1.10101715408560923160360372720,
1.10101715408560923160360372720, 2.03451897170989253356148732431, 3.64271201299328746371537786528, 5.28988083268399132293722298205, 6.46416894052052834655430003192, 7.38888700808618748687935596601, 7.955187968227007460564027584057, 8.899608505640944065032256426165, 9.476891964606973669092818124945, 10.29782179163196724125717137600