L(s) = 1 | + 2-s + 4-s − 1.41·5-s + 8-s − 3·9-s − 1.41·10-s − 2·11-s + 7.07·13-s + 16-s − 1.41·17-s − 3·18-s − 2.82·19-s − 1.41·20-s − 2·22-s − 6·23-s − 2.99·25-s + 7.07·26-s − 29-s + 1.41·31-s + 32-s − 1.41·34-s − 3·36-s − 4·37-s − 2.82·38-s − 1.41·40-s − 7.07·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 9-s − 0.447·10-s − 0.603·11-s + 1.96·13-s + 0.250·16-s − 0.342·17-s − 0.707·18-s − 0.648·19-s − 0.316·20-s − 0.426·22-s − 1.25·23-s − 0.599·25-s + 1.38·26-s − 0.185·29-s + 0.254·31-s + 0.176·32-s − 0.242·34-s − 0.5·36-s − 0.657·37-s − 0.458·38-s − 0.223·40-s − 1.10·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.391380315410183321566307985131, −7.75125707506951203302501292996, −6.68087583947254292067490665931, −5.98782726668821337417103714650, −5.44380380667879276791351701699, −4.26471953308040053037934590732, −3.71068413147365303744513046607, −2.84903477660818846975483656186, −1.70828781096723838430056037580, 0,
1.70828781096723838430056037580, 2.84903477660818846975483656186, 3.71068413147365303744513046607, 4.26471953308040053037934590732, 5.44380380667879276791351701699, 5.98782726668821337417103714650, 6.68087583947254292067490665931, 7.75125707506951203302501292996, 8.391380315410183321566307985131