L(s) = 1 | − 2·4-s − 5-s − 7-s + 2·11-s − 5·13-s + 4·16-s + 2·19-s + 2·20-s − 23-s + 25-s + 2·28-s + 8·29-s − 31-s + 35-s + 3·37-s − 7·41-s − 4·44-s + 47-s + 49-s + 10·52-s + 8·53-s − 2·55-s + 7·61-s − 8·64-s + 5·65-s + 16·67-s − 10·71-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s + 0.603·11-s − 1.38·13-s + 16-s + 0.458·19-s + 0.447·20-s − 0.208·23-s + 1/5·25-s + 0.377·28-s + 1.48·29-s − 0.179·31-s + 0.169·35-s + 0.493·37-s − 1.09·41-s − 0.603·44-s + 0.145·47-s + 1/7·49-s + 1.38·52-s + 1.09·53-s − 0.269·55-s + 0.896·61-s − 64-s + 0.620·65-s + 1.95·67-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.402611150\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.402611150\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.94554207437811, −13.32460622629907, −12.91187043700990, −12.32469964963852, −11.89636581667159, −11.71971794022635, −10.72802283724803, −10.31095838501694, −9.733741350035130, −9.490403621016053, −8.877547717762594, −8.326794528678977, −7.923309337172661, −7.273721306051566, −6.782992210881994, −6.247247564860200, −5.406891446164450, −5.044151323841513, −4.500199780739346, −3.917682642143421, −3.420262791880752, −2.748539075937403, −2.037474596247330, −0.9754547923066802, −0.4766009314775503,
0.4766009314775503, 0.9754547923066802, 2.037474596247330, 2.748539075937403, 3.420262791880752, 3.917682642143421, 4.500199780739346, 5.044151323841513, 5.406891446164450, 6.247247564860200, 6.782992210881994, 7.273721306051566, 7.923309337172661, 8.326794528678977, 8.877547717762594, 9.490403621016053, 9.733741350035130, 10.31095838501694, 10.72802283724803, 11.71971794022635, 11.89636581667159, 12.32469964963852, 12.91187043700990, 13.32460622629907, 13.94554207437811