| L(s) = 1 | + 1.47·3-s + 5-s − 7-s − 0.828·9-s + 4.43·11-s + 0.930·13-s + 1.47·15-s + 3.45·17-s − 7.28·19-s − 1.47·21-s + 6.12·23-s + 25-s − 5.64·27-s + 0.730·29-s + 8.04·31-s + 6.52·33-s − 35-s − 4.49·37-s + 1.37·39-s − 3.11·41-s + 43-s − 0.828·45-s + 4.02·47-s + 49-s + 5.08·51-s + 7.53·53-s + 4.43·55-s + ⋯ |
| L(s) = 1 | + 0.850·3-s + 0.447·5-s − 0.377·7-s − 0.276·9-s + 1.33·11-s + 0.258·13-s + 0.380·15-s + 0.836·17-s − 1.67·19-s − 0.321·21-s + 1.27·23-s + 0.200·25-s − 1.08·27-s + 0.135·29-s + 1.44·31-s + 1.13·33-s − 0.169·35-s − 0.738·37-s + 0.219·39-s − 0.486·41-s + 0.152·43-s − 0.123·45-s + 0.587·47-s + 0.142·49-s + 0.712·51-s + 1.03·53-s + 0.597·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.098516373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.098516373\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 - 0.930T + 13T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 + 7.28T + 19T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 - 0.730T + 29T^{2} \) |
| 31 | \( 1 - 8.04T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 - 3.95T + 59T^{2} \) |
| 61 | \( 1 - 8.42T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 - 0.640T + 73T^{2} \) |
| 79 | \( 1 - 7.71T + 79T^{2} \) |
| 83 | \( 1 - 6.09T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.414838428583935851978957998492, −7.33752608460273818594137502269, −6.61001701100918076197717188629, −6.12457774680921975228787142189, −5.24007107613881114001174979486, −4.23515349175189715931137809334, −3.55942887443449878354313746926, −2.80930207680068059012904258092, −1.98000923523882487811281534133, −0.911667581509630873612694761413,
0.911667581509630873612694761413, 1.98000923523882487811281534133, 2.80930207680068059012904258092, 3.55942887443449878354313746926, 4.23515349175189715931137809334, 5.24007107613881114001174979486, 6.12457774680921975228787142189, 6.61001701100918076197717188629, 7.33752608460273818594137502269, 8.414838428583935851978957998492
