L(s) = 1 | − 2.18·2-s + 2.76·4-s + 0.541·5-s + 0.936·7-s − 1.66·8-s − 1.18·10-s − 3.69·11-s − 4.89·13-s − 2.04·14-s − 1.89·16-s + 7.10·17-s − 1.09·19-s + 1.49·20-s + 8.07·22-s + 23-s − 4.70·25-s + 10.6·26-s + 2.58·28-s + 29-s − 5.91·31-s + 7.45·32-s − 15.5·34-s + 0.507·35-s + 10.3·37-s + 2.39·38-s − 0.901·40-s − 0.971·41-s + ⋯ |
L(s) = 1 | − 1.54·2-s + 1.38·4-s + 0.242·5-s + 0.354·7-s − 0.588·8-s − 0.373·10-s − 1.11·11-s − 1.35·13-s − 0.546·14-s − 0.473·16-s + 1.72·17-s − 0.251·19-s + 0.334·20-s + 1.72·22-s + 0.208·23-s − 0.941·25-s + 2.09·26-s + 0.489·28-s + 0.185·29-s − 1.06·31-s + 1.31·32-s − 2.66·34-s + 0.0857·35-s + 1.70·37-s + 0.387·38-s − 0.142·40-s − 0.151·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 5 | \( 1 - 0.541T + 5T^{2} \) |
| 7 | \( 1 - 0.936T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.971T + 41T^{2} \) |
| 43 | \( 1 + 1.94T + 43T^{2} \) |
| 47 | \( 1 - 6.84T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.09T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 + 18.7T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.916722067994951706706098420043, −7.41767024061333519662289857966, −6.62866131780273567064486617855, −5.52569220960674702136723889349, −5.12102524655708134889454759201, −3.98926371015296485521337493324, −2.72920800917577710150044210508, −2.16878627761105385578599435211, −1.09869798412030860395334704737, 0,
1.09869798412030860395334704737, 2.16878627761105385578599435211, 2.72920800917577710150044210508, 3.98926371015296485521337493324, 5.12102524655708134889454759201, 5.52569220960674702136723889349, 6.62866131780273567064486617855, 7.41767024061333519662289857966, 7.916722067994951706706098420043