| L(s) = 1 | + 1.84·3-s − 1.41·5-s + 0.765·7-s + 2.41·9-s − 0.765·11-s − 2.61·15-s − 1.84·19-s + 1.41·21-s + 1.00·25-s + 2.61·27-s − 1.41·33-s − 1.08·35-s + 1.41·37-s − 41-s − 3.41·45-s − 1.84·47-s − 0.414·49-s + 1.08·55-s − 3.41·57-s + 1.84·63-s − 0.765·67-s + 1.84·71-s + 1.41·73-s + 1.84·75-s − 0.585·77-s − 1.84·79-s + 2.41·81-s + ⋯ |
| L(s) = 1 | + 1.84·3-s − 1.41·5-s + 0.765·7-s + 2.41·9-s − 0.765·11-s − 2.61·15-s − 1.84·19-s + 1.41·21-s + 1.00·25-s + 2.61·27-s − 1.41·33-s − 1.08·35-s + 1.41·37-s − 41-s − 3.41·45-s − 1.84·47-s − 0.414·49-s + 1.08·55-s − 3.41·57-s + 1.84·63-s − 0.765·67-s + 1.84·71-s + 1.41·73-s + 1.84·75-s − 0.585·77-s − 1.84·79-s + 2.41·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.374354146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374354146\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.84T + T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - 0.765T + T^{2} \) |
| 11 | \( 1 + 0.765T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.84T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.84T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 0.765T + T^{2} \) |
| 71 | \( 1 - 1.84T + T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 + 1.84T + T^{2} \) |
| 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
−10.70596336240725264149232225268, −9.733007792262387254692434802912, −8.604321897841215865318537191356, −8.161834842290546298745261363336, −7.73360455698427098279862112553, −6.71176168031284065746505645065, −4.74899388908536537760678302633, −4.07733114932509869475177614089, −3.10784910675280961693792871803, −1.99525563756140697030623004922,
1.99525563756140697030623004922, 3.10784910675280961693792871803, 4.07733114932509869475177614089, 4.74899388908536537760678302633, 6.71176168031284065746505645065, 7.73360455698427098279862112553, 8.161834842290546298745261363336, 8.604321897841215865318537191356, 9.733007792262387254692434802912, 10.70596336240725264149232225268
