| L(s) = 1 | + 2.35·3-s + 3.68·5-s + 2.56·9-s + 11-s − 3.39·13-s + 8.68·15-s + 1.32·17-s − 4.71·19-s + 5.56·23-s + 8.56·25-s − 1.03·27-s + 2·29-s + 5.00·31-s + 2.35·33-s + 4.43·37-s − 8·39-s + 1.32·41-s + 10.2·43-s + 9.43·45-s − 1.32·47-s + 3.12·51-s − 2·53-s + 3.68·55-s − 11.1·57-s + 11.7·59-s − 13.4·61-s − 12.4·65-s + ⋯ |
| L(s) = 1 | + 1.36·3-s + 1.64·5-s + 0.853·9-s + 0.301·11-s − 0.940·13-s + 2.24·15-s + 0.321·17-s − 1.08·19-s + 1.15·23-s + 1.71·25-s − 0.198·27-s + 0.371·29-s + 0.899·31-s + 0.410·33-s + 0.729·37-s − 1.28·39-s + 0.206·41-s + 1.56·43-s + 1.40·45-s − 0.193·47-s + 0.437·51-s − 0.274·53-s + 0.496·55-s − 1.47·57-s + 1.53·59-s − 1.71·61-s − 1.54·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.043020543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.043020543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.00T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 1.32T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 + 3.80T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.85567255080355562847583427464, −7.13231044333591702047401096947, −6.41319992590843934516298727273, −5.79738789667582902165474414055, −4.93231729567469665954664756312, −4.25171470191777348627707641874, −3.15238955535132777896723128900, −2.52723541139063527929832355250, −2.08499909824521417437406314526, −1.06932176766277946982444878355,
1.06932176766277946982444878355, 2.08499909824521417437406314526, 2.52723541139063527929832355250, 3.15238955535132777896723128900, 4.25171470191777348627707641874, 4.93231729567469665954664756312, 5.79738789667582902165474414055, 6.41319992590843934516298727273, 7.13231044333591702047401096947, 7.85567255080355562847583427464
