| L(s) = 1 | + 4·11-s + 13-s − 8·17-s + 5·19-s + 4·23-s − 9·29-s − 4·31-s − 3·37-s − 5·41-s + 6·43-s + 5·47-s − 7·49-s + 5·53-s + 6·59-s + 4·61-s − 3·67-s − 7·71-s + 4·73-s − 79-s + 6·83-s + 6·89-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.277·13-s − 1.94·17-s + 1.14·19-s + 0.834·23-s − 1.67·29-s − 0.718·31-s − 0.493·37-s − 0.780·41-s + 0.914·43-s + 0.729·47-s − 49-s + 0.686·53-s + 0.781·59-s + 0.512·61-s − 0.366·67-s − 0.830·71-s + 0.468·73-s − 0.112·79-s + 0.658·83-s + 0.635·89-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.130280930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130280930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.39977775614815, −14.95422685713606, −14.39916221641515, −13.83026437544980, −13.21612069891785, −12.96556786769485, −12.07824815525353, −11.61813264799484, −11.09037100736868, −10.75940951608935, −9.811335986650645, −9.246635258517178, −8.952044464595942, −8.385659789031956, −7.369567065660737, −7.090936823389070, −6.469991275913687, −5.774629763130301, −5.152382644631880, −4.387749977180234, −3.799616228520196, −3.209707812555261, −2.193603771458857, −1.583026915048347, −0.5926783353546874,
0.5926783353546874, 1.583026915048347, 2.193603771458857, 3.209707812555261, 3.799616228520196, 4.387749977180234, 5.152382644631880, 5.774629763130301, 6.469991275913687, 7.090936823389070, 7.369567065660737, 8.385659789031956, 8.952044464595942, 9.246635258517178, 9.811335986650645, 10.75940951608935, 11.09037100736868, 11.61813264799484, 12.07824815525353, 12.96556786769485, 13.21612069891785, 13.83026437544980, 14.39916221641515, 14.95422685713606, 15.39977775614815
