| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 13-s − 15-s − 17-s − 19-s + 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s − 53-s − 57-s + 59-s + 61-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 13-s − 15-s − 17-s − 19-s + 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s − 53-s − 57-s + 59-s + 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.250227008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250227008\) |
| \(L(1)\) |
\(\approx\) |
\(1.274160921\) |
| \(L(1)\) |
\(\approx\) |
\(1.274160921\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 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
−30.79616438682801902202325065388, −29.95446736859443145816977854585, −28.16201637197724422147255280210, −27.32562042436623326985861774449, −26.4612327489981952691005663849, −25.34842235289871858803268534422, −24.16327435238130761326533452835, −23.49120404241694356171217010745, −21.86276693106788534010884661714, −20.71509657147769274301627726467, −19.95810511784958920726513661051, −18.87465874783405690951370408588, −17.83721808852249725079656196146, −16.08405286828134801688820172170, −15.21444027545195151315046906436, −14.25379827623381915783058812290, −13.03850016615324348703912328502, −11.6412117342240508941881936625, −10.51354001035745447006072907722, −8.65692780189773129117247054146, −8.18927219855589579296726046868, −6.80866416106449003054026860076, −4.625742714333304902959048108232, −3.6067820083115103544047058571, −1.86469995987961884344946415760,
1.86469995987961884344946415760, 3.6067820083115103544047058571, 4.625742714333304902959048108232, 6.80866416106449003054026860076, 8.18927219855589579296726046868, 8.65692780189773129117247054146, 10.51354001035745447006072907722, 11.6412117342240508941881936625, 13.03850016615324348703912328502, 14.25379827623381915783058812290, 15.21444027545195151315046906436, 16.08405286828134801688820172170, 17.83721808852249725079656196146, 18.87465874783405690951370408588, 19.95810511784958920726513661051, 20.71509657147769274301627726467, 21.86276693106788534010884661714, 23.49120404241694356171217010745, 24.16327435238130761326533452835, 25.34842235289871858803268534422, 26.4612327489981952691005663849, 27.32562042436623326985861774449, 28.16201637197724422147255280210, 29.95446736859443145816977854585, 30.79616438682801902202325065388
