| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 21-s + 23-s + 25-s + 27-s + 29-s − 31-s − 33-s − 35-s + 37-s + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s + 53-s + 55-s + 59-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 21-s + 23-s + 25-s + 27-s + 29-s − 31-s − 33-s − 35-s + 37-s + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s + 53-s + 55-s + 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.547011170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.547011170\) |
| \(L(1)\) |
\(\approx\) |
\(1.528900874\) |
| \(L(1)\) |
\(\approx\) |
\(1.528900874\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 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
−27.45887285573937389820892798541, −26.94809649814633933966714367866, −25.85024417749752603888994601316, −24.95406852489626898469612227179, −23.72252832878027149694686841421, −23.335080669753099723586315482, −21.51870268947889990682587448961, −20.77525963658662156807406933432, −20.01976512568988203097134503495, −18.77260169016316670869643134353, −18.2645953095371647299663582579, −16.52909859818801048029930065168, −15.4792864110982358339660426088, −14.79826939888833507246297202049, −13.69400838589107345023346018473, −12.5982080877174889675810617196, −11.331986857680979264308302444429, −10.31849529825159021449394279600, −8.69716745995560468494862953100, −8.06630160341600837143926917835, −7.174690552612637897130104196306, −5.17808792097857683595651995091, −3.97177712649849110830621889787, −2.84505871867905888360710213691, −1.191133512716638333409087237597,
1.191133512716638333409087237597, 2.84505871867905888360710213691, 3.97177712649849110830621889787, 5.17808792097857683595651995091, 7.174690552612637897130104196306, 8.06630160341600837143926917835, 8.69716745995560468494862953100, 10.31849529825159021449394279600, 11.331986857680979264308302444429, 12.5982080877174889675810617196, 13.69400838589107345023346018473, 14.79826939888833507246297202049, 15.4792864110982358339660426088, 16.52909859818801048029930065168, 18.2645953095371647299663582579, 18.77260169016316670869643134353, 20.01976512568988203097134503495, 20.77525963658662156807406933432, 21.51870268947889990682587448961, 23.335080669753099723586315482, 23.72252832878027149694686841421, 24.95406852489626898469612227179, 25.85024417749752603888994601316, 26.94809649814633933966714367866, 27.45887285573937389820892798541
