L(s) = 1 | − 2·3-s − 7-s + 9-s + 6·17-s − 6·19-s + 2·21-s + 23-s − 5·25-s + 4·27-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s + 8·43-s − 8·47-s + 49-s − 12·51-s − 2·53-s + 12·57-s − 6·59-s − 63-s − 12·67-s − 2·69-s − 8·71-s + 6·73-s + 10·75-s + 16·79-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.45·17-s − 1.37·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 1.68·51-s − 0.274·53-s + 1.58·57-s − 0.781·59-s − 0.125·63-s − 1.46·67-s − 0.240·69-s − 0.949·71-s + 0.702·73-s + 1.15·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.088472215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.088472215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.24549133871715, −12.62765319243998, −12.23135181749915, −11.99480836062590, −11.45494786621351, −10.91204134592195, −10.39825167829313, −10.20729808405112, −9.539592736791400, −9.098816000083687, −8.312662076138275, −8.014357971090868, −7.467181575133010, −6.679445776619944, −6.325521994376455, −6.019756464443974, −5.450383230708389, −4.885205312119928, −4.370489399341359, −3.864669135992683, −2.972550353289681, −2.691685071782665, −1.703007528722514, −1.017494718173147, −0.3882928899538985,
0.3882928899538985, 1.017494718173147, 1.703007528722514, 2.691685071782665, 2.972550353289681, 3.864669135992683, 4.370489399341359, 4.885205312119928, 5.450383230708389, 6.019756464443974, 6.325521994376455, 6.679445776619944, 7.467181575133010, 8.014357971090868, 8.312662076138275, 9.098816000083687, 9.539592736791400, 10.20729808405112, 10.39825167829313, 10.91204134592195, 11.45494786621351, 11.99480836062590, 12.23135181749915, 12.62765319243998, 13.24549133871715