L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 7-s + 9-s − 2·11-s + 2·12-s − 13-s − 2·14-s − 4·16-s + 4·17-s + 2·18-s + 3·19-s − 21-s − 4·22-s + 9·23-s − 2·26-s + 27-s − 2·28-s − 29-s − 5·31-s − 8·32-s − 2·33-s + 8·34-s + 2·36-s + 8·37-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 0.277·13-s − 0.534·14-s − 16-s + 0.970·17-s + 0.471·18-s + 0.688·19-s − 0.218·21-s − 0.852·22-s + 1.87·23-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.185·29-s − 0.898·31-s − 1.41·32-s − 0.348·33-s + 1.37·34-s + 1/3·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.199018686\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.199018686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.60768388532081639670239930241, −7.31434781449891418486826893676, −6.40731480360730782128027419060, −5.58094419807793489744167477468, −5.16918054563849842901866650235, −4.32089518432251534600417132641, −3.56818021237328646863954713878, −2.92920235488280143664457673115, −2.38138644519741461749494273823, −0.922341013465717185485052242822,
0.922341013465717185485052242822, 2.38138644519741461749494273823, 2.92920235488280143664457673115, 3.56818021237328646863954713878, 4.32089518432251534600417132641, 5.16918054563849842901866650235, 5.58094419807793489744167477468, 6.40731480360730782128027419060, 7.31434781449891418486826893676, 7.60768388532081639670239930241