L(s) = 1 | + 3·5-s + 7-s − 5·11-s − 7·17-s − 3·19-s − 3·23-s + 4·25-s − 7·29-s + 4·31-s + 3·35-s − 11·37-s − 4·41-s + 43-s − 8·47-s + 49-s − 2·53-s − 15·55-s − 6·59-s + 13·61-s − 4·67-s + 6·71-s − 73-s − 5·77-s + 8·79-s − 14·83-s − 21·85-s − 9·95-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s − 1.50·11-s − 1.69·17-s − 0.688·19-s − 0.625·23-s + 4/5·25-s − 1.29·29-s + 0.718·31-s + 0.507·35-s − 1.80·37-s − 0.624·41-s + 0.152·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 2.02·55-s − 0.781·59-s + 1.66·61-s − 0.488·67-s + 0.712·71-s − 0.117·73-s − 0.569·77-s + 0.900·79-s − 1.53·83-s − 2.27·85-s − 0.923·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.063755514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.063755514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.84279654482526, −13.40412887025869, −13.03996943509311, −12.70097452740961, −11.96581457273029, −11.27824575326138, −10.90028958758394, −10.41053756387510, −9.980934277415854, −9.545474836045102, −8.749582149700619, −8.563100356935313, −7.904818438593899, −7.277552656469586, −6.643584822038961, −6.249787008824148, −5.587226628092580, −5.151611064594917, −4.708307002696286, −3.989295431877868, −3.184326696248007, −2.410178734076865, −2.042032573892057, −1.619426707091632, −0.2983621205161657,
0.2983621205161657, 1.619426707091632, 2.042032573892057, 2.410178734076865, 3.184326696248007, 3.989295431877868, 4.708307002696286, 5.151611064594917, 5.587226628092580, 6.249787008824148, 6.643584822038961, 7.277552656469586, 7.904818438593899, 8.563100356935313, 8.749582149700619, 9.545474836045102, 9.980934277415854, 10.41053756387510, 10.90028958758394, 11.27824575326138, 11.96581457273029, 12.70097452740961, 13.03996943509311, 13.40412887025869, 13.84279654482526