| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.472594897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472594897\) |
| \(L(1)\) |
\(\approx\) |
\(1.297484958\) |
| \(L(1)\) |
\(\approx\) |
\(1.297484958\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 257 | \( 1 \) |
| 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 | \( 1 + T \) |
| 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
−25.650128456591231597713190652670, −24.831054771113939123139270496156, −23.611316282951626138759105314060, −23.1400033561315948485872855061, −22.60843485848774462887405293275, −21.613000933695876187726390905201, −20.650430613811411601947687400066, −19.348897853151617711032014744836, −18.948102450661743937630062668707, −17.17237948322151093109957014966, −16.39890987122083635337593929123, −15.73496087881392604844841350544, −14.84135362143036127741758887490, −13.46687655170719242393196554773, −12.51263317173369393286497599899, −11.89449767695401582652288383259, −11.02660225371539141976394525632, −10.04631836159841083066628295658, −8.33603387115014370767073788385, −6.75714686474172771126487300581, −6.52028120064304478429608013131, −5.145405538542661124592171621607, −4.01982097512319808139344964343, −3.27978550447527314018450426103, −1.17548352834327260240532392652,
1.17548352834327260240532392652, 3.27978550447527314018450426103, 4.01982097512319808139344964343, 5.145405538542661124592171621607, 6.52028120064304478429608013131, 6.75714686474172771126487300581, 8.33603387115014370767073788385, 10.04631836159841083066628295658, 11.02660225371539141976394525632, 11.89449767695401582652288383259, 12.51263317173369393286497599899, 13.46687655170719242393196554773, 14.84135362143036127741758887490, 15.73496087881392604844841350544, 16.39890987122083635337593929123, 17.17237948322151093109957014966, 18.948102450661743937630062668707, 19.348897853151617711032014744836, 20.650430613811411601947687400066, 21.613000933695876187726390905201, 22.60843485848774462887405293275, 23.1400033561315948485872855061, 23.611316282951626138759105314060, 24.831054771113939123139270496156, 25.650128456591231597713190652670
