L(s) = 1 | + 0.505·3-s − 3.97·5-s − 1.36·7-s − 2.74·9-s − 4.72·11-s − 3.07·13-s − 2.00·15-s − 2.19·17-s − 5.12·19-s − 0.690·21-s − 5.63·23-s + 10.8·25-s − 2.90·27-s + 8.38·29-s − 5.97·31-s − 2.38·33-s + 5.43·35-s − 0.0197·37-s − 1.55·39-s − 8.14·41-s − 1.90·43-s + 10.9·45-s + 1.09·47-s − 5.12·49-s − 1.10·51-s + 7.79·53-s + 18.7·55-s + ⋯ |
L(s) = 1 | + 0.291·3-s − 1.77·5-s − 0.516·7-s − 0.914·9-s − 1.42·11-s − 0.852·13-s − 0.518·15-s − 0.532·17-s − 1.17·19-s − 0.150·21-s − 1.17·23-s + 2.16·25-s − 0.558·27-s + 1.55·29-s − 1.07·31-s − 0.415·33-s + 0.919·35-s − 0.00324·37-s − 0.248·39-s − 1.27·41-s − 0.290·43-s + 1.62·45-s + 0.159·47-s − 0.732·49-s − 0.155·51-s + 1.07·53-s + 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05612054933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05612054933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 0.505T + 3T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 + 5.97T + 31T^{2} \) |
| 37 | \( 1 + 0.0197T + 37T^{2} \) |
| 41 | \( 1 + 8.14T + 41T^{2} \) |
| 43 | \( 1 + 1.90T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 4.79T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 0.906T + 97T^{2} \) |
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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
−8.425432386831450755313704665163, −7.82418762462146709845279703410, −7.16901315802953909642439733851, −6.37588658006153263683995379176, −5.32241208597903735350302659552, −4.57624672898882976090056475311, −3.79618790858218832729083586028, −2.97548235291208678613950461731, −2.31359206522672441955347062819, −0.12136815475497691553484153397,
0.12136815475497691553484153397, 2.31359206522672441955347062819, 2.97548235291208678613950461731, 3.79618790858218832729083586028, 4.57624672898882976090056475311, 5.32241208597903735350302659552, 6.37588658006153263683995379176, 7.16901315802953909642439733851, 7.82418762462146709845279703410, 8.425432386831450755313704665163