L(s) = 1 | + 1.61·5-s − 0.236·7-s + 4.23·13-s − 0.145·17-s + 6.85·19-s + 5·23-s − 2.38·25-s + 2·29-s + 3.38·31-s − 0.381·35-s + 2.23·37-s − 8.70·41-s + 2.52·43-s − 9.56·47-s − 6.94·49-s − 6.61·53-s − 11.7·59-s + 10.8·61-s + 6.85·65-s − 4.38·67-s + 14.5·71-s + 15.7·73-s + 8.70·79-s − 1.76·83-s − 0.236·85-s + 5.18·89-s − 91-s + ⋯ |
L(s) = 1 | + 0.723·5-s − 0.0892·7-s + 1.17·13-s − 0.0353·17-s + 1.57·19-s + 1.04·23-s − 0.476·25-s + 0.371·29-s + 0.607·31-s − 0.0645·35-s + 0.367·37-s − 1.35·41-s + 0.385·43-s − 1.39·47-s − 0.992·49-s − 0.909·53-s − 1.53·59-s + 1.38·61-s + 0.850·65-s − 0.535·67-s + 1.72·71-s + 1.83·73-s + 0.979·79-s − 0.193·83-s − 0.0256·85-s + 0.549·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.529025912\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.529025912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 0.145T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 8.70T + 79T^{2} \) |
| 83 | \( 1 + 1.76T + 83T^{2} \) |
| 89 | \( 1 - 5.18T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.301765573339502091598945322883, −7.77366353106139019759806425992, −6.67808049581824487170938694666, −6.30940568211567255671151212219, −5.36517591829451832977396274792, −4.83964504894207689518538279979, −3.60225592656979273761854875280, −3.03659525366274430065415222892, −1.83668388883843981735394852710, −0.954693359886716852724160711270,
0.954693359886716852724160711270, 1.83668388883843981735394852710, 3.03659525366274430065415222892, 3.60225592656979273761854875280, 4.83964504894207689518538279979, 5.36517591829451832977396274792, 6.30940568211567255671151212219, 6.67808049581824487170938694666, 7.77366353106139019759806425992, 8.301765573339502091598945322883