| L(s) = 1 | − 2.70·5-s + 3.36·7-s + 5.17·11-s + 13-s − 6.70·17-s + 5.17·19-s + 2.29·25-s + 2·29-s + 8.80·31-s − 9.08·35-s + 2.70·37-s − 3.40·41-s − 8.53·43-s − 3.36·47-s + 4.29·49-s + 11.4·53-s − 13.9·55-s − 2.08·59-s − 3.40·61-s − 2.70·65-s + 12.4·67-s + 10.6·71-s − 6·73-s + 17.4·77-s − 3.09·79-s − 1.54·83-s + 18.1·85-s + ⋯ |
| L(s) = 1 | − 1.20·5-s + 1.27·7-s + 1.56·11-s + 0.277·13-s − 1.62·17-s + 1.18·19-s + 0.459·25-s + 0.371·29-s + 1.58·31-s − 1.53·35-s + 0.444·37-s − 0.531·41-s − 1.30·43-s − 0.490·47-s + 0.614·49-s + 1.56·53-s − 1.88·55-s − 0.271·59-s − 0.435·61-s − 0.335·65-s + 1.52·67-s + 1.26·71-s − 0.702·73-s + 1.98·77-s − 0.347·79-s − 0.169·83-s + 1.96·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.937428049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937428049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 + 3.36T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.318339275042352600767028822053, −8.023965718941274779584677145985, −6.94037757065210600290043927801, −6.59665261054919840049895034955, −5.35525139582793946644691887567, −4.45102641330012482890627829081, −4.13108447960093097044784001931, −3.13376265771716670281498363812, −1.83477529151229415602370966791, −0.857161014460704608857579854590,
0.857161014460704608857579854590, 1.83477529151229415602370966791, 3.13376265771716670281498363812, 4.13108447960093097044784001931, 4.45102641330012482890627829081, 5.35525139582793946644691887567, 6.59665261054919840049895034955, 6.94037757065210600290043927801, 8.023965718941274779584677145985, 8.318339275042352600767028822053
