| L(s) = 1 | + 3-s − 0.0546·5-s − 3.75·7-s + 9-s − 3.94·11-s + 6.73·13-s − 0.0546·15-s + 7.10·17-s + 6.19·19-s − 3.75·21-s − 0.493·23-s − 4.99·25-s + 27-s − 5.62·29-s − 3.17·31-s − 3.94·33-s + 0.204·35-s + 0.462·37-s + 6.73·39-s − 8.07·41-s − 7.28·43-s − 0.0546·45-s + 9.82·47-s + 7.08·49-s + 7.10·51-s + 7.30·53-s + 0.215·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0244·5-s − 1.41·7-s + 0.333·9-s − 1.18·11-s + 1.86·13-s − 0.0141·15-s + 1.72·17-s + 1.42·19-s − 0.818·21-s − 0.102·23-s − 0.999·25-s + 0.192·27-s − 1.04·29-s − 0.570·31-s − 0.686·33-s + 0.0346·35-s + 0.0761·37-s + 1.07·39-s − 1.26·41-s − 1.11·43-s − 0.00814·45-s + 1.43·47-s + 1.01·49-s + 0.994·51-s + 1.00·53-s + 0.0290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143542798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143542798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.0546T + 5T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 6.73T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 + 0.493T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 0.462T + 37T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 + 7.28T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 - 2.42T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 - 6.05T + 73T^{2} \) |
| 79 | \( 1 + 0.493T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.81000909295769074159009824275, −7.30730089171959135163958954015, −6.43683609305873247729058688683, −5.63102961726747665666802970598, −5.36841416226772266816583738108, −3.82518939338907581810869005699, −3.48638428238791277805377025358, −2.96685577970860030833592583843, −1.79494357417120542110223736711, −0.70729668570476190216303207033,
0.70729668570476190216303207033, 1.79494357417120542110223736711, 2.96685577970860030833592583843, 3.48638428238791277805377025358, 3.82518939338907581810869005699, 5.36841416226772266816583738108, 5.63102961726747665666802970598, 6.43683609305873247729058688683, 7.30730089171959135163958954015, 7.81000909295769074159009824275
