| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s − 29-s + 30-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s − 29-s + 30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3839 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3839 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(13.35957055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.35957055\) |
| \(L(1)\) |
\(\approx\) |
\(4.157714547\) |
| \(L(1)\) |
\(\approx\) |
\(4.157714547\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 349 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 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
−18.52582222092218254691178846002, −17.64176729419868650020166481754, −17.02968866813674780729430386167, −16.12880516275970606954134405104, −15.27759823316731261717099285324, −14.88802463435267265376582262931, −14.2227431728088208467263866692, −13.59990976769331743007598626023, −13.13258711892186998306490966316, −12.61713701205020658891156956149, −11.35091325551045789458875649884, −10.94196261363745211857508514216, −10.19109686705875088876071895485, −9.22940073592974198100369562397, −8.5434116367598537071426205029, −7.91334839474697577560713181769, −6.90662887579088076156129832012, −6.3866906455745537867897646807, −5.508335605797711616380881908096, −4.6283410110115792142555961235, −4.18538767072720146701457480373, −3.14378525423230946406280825622, −2.42060728008388880452727907062, −1.76621490226852325956075813199, −1.17101217251290383884852129155,
1.17101217251290383884852129155, 1.76621490226852325956075813199, 2.42060728008388880452727907062, 3.14378525423230946406280825622, 4.18538767072720146701457480373, 4.6283410110115792142555961235, 5.508335605797711616380881908096, 6.3866906455745537867897646807, 6.90662887579088076156129832012, 7.91334839474697577560713181769, 8.5434116367598537071426205029, 9.22940073592974198100369562397, 10.19109686705875088876071895485, 10.94196261363745211857508514216, 11.35091325551045789458875649884, 12.61713701205020658891156956149, 13.13258711892186998306490966316, 13.59990976769331743007598626023, 14.2227431728088208467263866692, 14.88802463435267265376582262931, 15.27759823316731261717099285324, 16.12880516275970606954134405104, 17.02968866813674780729430386167, 17.64176729419868650020166481754, 18.52582222092218254691178846002
