| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.700360921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.700360921\) |
| \(L(1)\) |
\(\approx\) |
\(1.767283942\) |
| \(L(1)\) |
\(\approx\) |
\(1.767283942\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 79 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.68358034974079237310677886471, −29.7195251294510546157497565014, −28.94700751018617629623641340055, −28.233448325375724699591778602803, −26.375487460715074987144174317832, −25.14092645174356715836722426441, −24.365129416210414050644238862157, −22.90071311658847917527897749780, −22.40065830760721059643677620366, −21.49825758241748941086239053326, −20.28606729690541477249467568433, −18.77920366376857385493639433272, −17.32968088359014878143636459701, −16.4370615355295614803399695706, −15.37837822853404269695340278458, −13.70126095905041503764610577051, −13.0264037453284194984112464751, −11.769771030938999370368512198739, −10.65636041802132783286354515039, −9.38933148107311761708893358496, −6.762211042998195975623374744878, −6.25702467824181093607360901198, −5.019924093927836233834444108, −3.44180406603687118064045090486, −1.466837987990898739497210662567,
1.466837987990898739497210662567, 3.44180406603687118064045090486, 5.019924093927836233834444108, 6.25702467824181093607360901198, 6.762211042998195975623374744878, 9.38933148107311761708893358496, 10.65636041802132783286354515039, 11.769771030938999370368512198739, 13.0264037453284194984112464751, 13.70126095905041503764610577051, 15.37837822853404269695340278458, 16.4370615355295614803399695706, 17.32968088359014878143636459701, 18.77920366376857385493639433272, 20.28606729690541477249467568433, 21.49825758241748941086239053326, 22.40065830760721059643677620366, 22.90071311658847917527897749780, 24.365129416210414050644238862157, 25.14092645174356715836722426441, 26.375487460715074987144174317832, 28.233448325375724699591778602803, 28.94700751018617629623641340055, 29.7195251294510546157497565014, 30.68358034974079237310677886471
