L(s) = 1 | + 1.38·3-s + 1.12·5-s + 1.87·7-s − 1.08·9-s − 4.20·11-s − 2.09·13-s + 1.55·15-s + 0.221·17-s − 2.15·19-s + 2.60·21-s + 5.86·23-s − 3.74·25-s − 5.65·27-s − 1.70·29-s − 1.13·31-s − 5.81·33-s + 2.10·35-s − 4.68·37-s − 2.90·39-s + 7.45·41-s − 5.19·43-s − 1.21·45-s − 11.1·47-s − 3.46·49-s + 0.307·51-s + 4.16·53-s − 4.71·55-s + ⋯ |
L(s) = 1 | + 0.799·3-s + 0.501·5-s + 0.710·7-s − 0.361·9-s − 1.26·11-s − 0.582·13-s + 0.400·15-s + 0.0538·17-s − 0.495·19-s + 0.567·21-s + 1.22·23-s − 0.748·25-s − 1.08·27-s − 0.315·29-s − 0.204·31-s − 1.01·33-s + 0.355·35-s − 0.770·37-s − 0.465·39-s + 1.16·41-s − 0.792·43-s − 0.181·45-s − 1.62·47-s − 0.495·49-s + 0.0430·51-s + 0.571·53-s − 0.635·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 1.38T + 3T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 - 0.221T + 17T^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 - 7.45T + 41T^{2} \) |
| 43 | \( 1 + 5.19T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.16T + 53T^{2} \) |
| 59 | \( 1 + 4.23T + 59T^{2} \) |
| 61 | \( 1 - 9.09T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.84553073822833160651670609659, −7.24343097734715149645002268878, −6.29849369795151849002285798746, −5.31420312617603315559352024033, −5.08064596280836144618833493067, −3.97003838830949219161450880094, −2.99063366071647815098905742758, −2.40725173482788742117879664356, −1.63001780590450911782390505332, 0,
1.63001780590450911782390505332, 2.40725173482788742117879664356, 2.99063366071647815098905742758, 3.97003838830949219161450880094, 5.08064596280836144618833493067, 5.31420312617603315559352024033, 6.29849369795151849002285798746, 7.24343097734715149645002268878, 7.84553073822833160651670609659