| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.208631876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208631876\) |
| \(L(1)\) |
\(\approx\) |
\(1.692740092\) |
| \(L(1)\) |
\(\approx\) |
\(1.692740092\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| 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 | \( 1 - T \) |
| 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
−36.67685000147278202904180240027, −34.73525141826695901662244337819, −33.72763908065133135528759026996, −33.21474384193483545413746361975, −31.63934929805853763413028640302, −30.22023902961814429222295335058, −29.23477470645741101301735856374, −28.3374356036886045704236376649, −26.4601535274486392459102175141, −24.59995978410064295214080086541, −24.011330598375603608659460143202, −22.42070185559805391300678577861, −21.6070391912365273178177076636, −20.49918501249093324848331385672, −18.1687248306068221894848784442, −17.09514996166819244174822299264, −15.60301752682389109688066981158, −14.05885519779649141901602534311, −12.77142050335587766094724998350, −11.39258383036663027678688446981, −10.18385049058425751554958684335, −7.33591057329007776327141175518, −5.68644602105895428746100540523, −4.78968470194607603822353764529, −2.03498242395270329177915345847,
2.03498242395270329177915345847, 4.78968470194607603822353764529, 5.68644602105895428746100540523, 7.33591057329007776327141175518, 10.18385049058425751554958684335, 11.39258383036663027678688446981, 12.77142050335587766094724998350, 14.05885519779649141901602534311, 15.60301752682389109688066981158, 17.09514996166819244174822299264, 18.1687248306068221894848784442, 20.49918501249093324848331385672, 21.6070391912365273178177076636, 22.42070185559805391300678577861, 24.011330598375603608659460143202, 24.59995978410064295214080086541, 26.4601535274486392459102175141, 28.3374356036886045704236376649, 29.23477470645741101301735856374, 30.22023902961814429222295335058, 31.63934929805853763413028640302, 33.21474384193483545413746361975, 33.72763908065133135528759026996, 34.73525141826695901662244337819, 36.67685000147278202904180240027
