| L(s) = 1 | − 2.15·2-s − 0.229·3-s + 2.62·4-s + 4.13·5-s + 0.493·6-s − 0.301·7-s − 1.33·8-s − 2.94·9-s − 8.88·10-s − 5.38·11-s − 0.601·12-s − 0.811·13-s + 0.648·14-s − 0.948·15-s − 2.36·16-s + 17-s + 6.33·18-s + 2.69·19-s + 10.8·20-s + 0.0691·21-s + 11.5·22-s − 0.617·23-s + 0.307·24-s + 12.0·25-s + 1.74·26-s + 1.36·27-s − 0.790·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.132·3-s + 1.31·4-s + 1.84·5-s + 0.201·6-s − 0.113·7-s − 0.473·8-s − 0.982·9-s − 2.81·10-s − 1.62·11-s − 0.173·12-s − 0.224·13-s + 0.173·14-s − 0.244·15-s − 0.591·16-s + 0.242·17-s + 1.49·18-s + 0.617·19-s + 2.42·20-s + 0.0150·21-s + 2.46·22-s − 0.128·23-s + 0.0627·24-s + 2.41·25-s + 0.342·26-s + 0.262·27-s − 0.149·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8798281005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8798281005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 0.229T + 3T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 7 | \( 1 + 0.301T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 + 0.811T + 13T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 + 0.617T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 0.966T + 31T^{2} \) |
| 37 | \( 1 + 8.98T + 37T^{2} \) |
| 41 | \( 1 + 1.11T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 0.820T + 47T^{2} \) |
| 53 | \( 1 - 2.60T + 53T^{2} \) |
| 59 | \( 1 - 2.03T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 + 6.52T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 4.58T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.231632671749315559204795684227, −7.57046577340731808089953992647, −6.77544560618913842902453472894, −6.01224236551170447625483297163, −5.41929940071828566363856279170, −4.85803557712184657998044604352, −3.02558915504257749493387243456, −2.49287035874423420681504022161, −1.74165918495231426567007703341, −0.61401470089880109372778940531,
0.61401470089880109372778940531, 1.74165918495231426567007703341, 2.49287035874423420681504022161, 3.02558915504257749493387243456, 4.85803557712184657998044604352, 5.41929940071828566363856279170, 6.01224236551170447625483297163, 6.77544560618913842902453472894, 7.57046577340731808089953992647, 8.231632671749315559204795684227
