| L(s) = 1 | − 1.61·3-s + 4-s + 7-s + 1.61·9-s − 1.61·11-s − 1.61·12-s + 0.618·13-s + 16-s + 0.618·17-s − 1.61·21-s − 27-s + 28-s + 0.618·29-s + 2.61·33-s + 1.61·36-s − 1.00·39-s − 1.61·44-s + 2·47-s − 1.61·48-s + 49-s − 1.00·51-s + 0.618·52-s + 1.61·63-s + 64-s + 0.618·68-s + 0.618·71-s − 1.61·73-s + ⋯ |
| L(s) = 1 | − 1.61·3-s + 4-s + 7-s + 1.61·9-s − 1.61·11-s − 1.61·12-s + 0.618·13-s + 16-s + 0.618·17-s − 1.61·21-s − 27-s + 28-s + 0.618·29-s + 2.61·33-s + 1.61·36-s − 1.00·39-s − 1.61·44-s + 2·47-s − 1.61·48-s + 49-s − 1.00·51-s + 0.618·52-s + 1.61·63-s + 64-s + 0.618·68-s + 0.618·71-s − 1.61·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8153147333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8153147333\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 2T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 + 1.61T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−10.67804818250168426305407644334, −10.05994866857954715174119141180, −8.401137027450827670044559940443, −7.60803775021938387347644807084, −6.87407769194828373885813959353, −5.68675610253840970382126690119, −5.50189381042325431012807069944, −4.35190213302483704863429043956, −2.69611514254397000178644793630, −1.29464234200991065655870609094,
1.29464234200991065655870609094, 2.69611514254397000178644793630, 4.35190213302483704863429043956, 5.50189381042325431012807069944, 5.68675610253840970382126690119, 6.87407769194828373885813959353, 7.60803775021938387347644807084, 8.401137027450827670044559940443, 10.05994866857954715174119141180, 10.67804818250168426305407644334
