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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.795080609\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.795080609\) |
\(L(1)\) |
\(\approx\) |
\(2.081935563\) |
\(L(1)\) |
\(\approx\) |
\(2.081935563\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3089 | \( 1 \) |
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
−18.98649425919922027383647560632, −18.004199620609539367537306367470, −17.44351822324532954567558213817, −17.02493818469749755894759255905, −16.27950932735102271842304995026, −15.37364308290442122436725667941, −14.764347829619427936122611578206, −13.88570998871122028088828843038, −13.63144062534870326553588677240, −12.51959333312883130644871120369, −11.98085165623017008934874224365, −11.45913236734416915541628090527, −10.69307990958868372900884136291, −10.01160083113616235693446871198, −9.24187974306513054710002180997, −8.02767727997180390547686379038, −6.97356602603096927401170078286, −6.72920960020158238382711565522, −5.52306668304908549455122699397, −5.438708976340597120582302272595, −4.47067187451388179389619871835, −3.90957553692704579906895026368, −2.47236215421442630510624015700, −1.8512691452178303295978326314, −1.08123634509602035847304545739,
1.08123634509602035847304545739, 1.8512691452178303295978326314, 2.47236215421442630510624015700, 3.90957553692704579906895026368, 4.47067187451388179389619871835, 5.438708976340597120582302272595, 5.52306668304908549455122699397, 6.72920960020158238382711565522, 6.97356602603096927401170078286, 8.02767727997180390547686379038, 9.24187974306513054710002180997, 10.01160083113616235693446871198, 10.69307990958868372900884136291, 11.45913236734416915541628090527, 11.98085165623017008934874224365, 12.51959333312883130644871120369, 13.63144062534870326553588677240, 13.88570998871122028088828843038, 14.764347829619427936122611578206, 15.37364308290442122436725667941, 16.27950932735102271842304995026, 17.02493818469749755894759255905, 17.44351822324532954567558213817, 18.004199620609539367537306367470, 18.98649425919922027383647560632