L(s) = 1 | + 2·11-s − 2·13-s + 6·17-s + 8·19-s − 4·23-s − 8·29-s − 10·37-s + 2·41-s + 12·43-s − 10·53-s + 6·59-s − 2·61-s + 8·67-s − 4·71-s − 4·73-s + 8·79-s − 4·83-s + 6·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.603·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.834·23-s − 1.48·29-s − 1.64·37-s + 0.312·41-s + 1.82·43-s − 1.37·53-s + 0.781·59-s − 0.256·61-s + 0.977·67-s − 0.474·71-s − 0.468·73-s + 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.572821633\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.572821633\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.22257125584258, −12.50175237056076, −12.20189716948216, −11.94831646102844, −11.28170563558984, −10.92647123394346, −10.23123845594726, −9.741800400500758, −9.496659576301551, −9.042958685410436, −8.306243080592457, −7.810936516998147, −7.311974375723404, −7.161495052344370, −6.258612052278679, −5.769586462110108, −5.361162021021669, −4.922805170631118, −4.092095236503159, −3.618245563005240, −3.215859097658340, −2.507040848008916, −1.744535392384292, −1.231443735012665, −0.4851645464945056,
0.4851645464945056, 1.231443735012665, 1.744535392384292, 2.507040848008916, 3.215859097658340, 3.618245563005240, 4.092095236503159, 4.922805170631118, 5.361162021021669, 5.769586462110108, 6.258612052278679, 7.161495052344370, 7.311974375723404, 7.810936516998147, 8.306243080592457, 9.042958685410436, 9.496659576301551, 9.741800400500758, 10.23123845594726, 10.92647123394346, 11.28170563558984, 11.94831646102844, 12.20189716948216, 12.50175237056076, 13.22257125584258