L(s) = 1 | − 1.21·3-s + 3.23·5-s + 7-s − 1.51·9-s + 11-s − 13-s − 3.93·15-s − 3.51·17-s − 5.89·19-s − 1.21·21-s − 0.0895·23-s + 5.44·25-s + 5.50·27-s − 5.65·29-s + 4.09·31-s − 1.21·33-s + 3.23·35-s + 6.80·37-s + 1.21·39-s + 10.2·41-s + 5.55·43-s − 4.89·45-s + 7.36·47-s + 49-s + 4.27·51-s + 11.6·53-s + 3.23·55-s + ⋯ |
L(s) = 1 | − 0.703·3-s + 1.44·5-s + 0.377·7-s − 0.505·9-s + 0.301·11-s − 0.277·13-s − 1.01·15-s − 0.852·17-s − 1.35·19-s − 0.265·21-s − 0.0186·23-s + 1.08·25-s + 1.05·27-s − 1.04·29-s + 0.735·31-s − 0.212·33-s + 0.546·35-s + 1.11·37-s + 0.195·39-s + 1.59·41-s + 0.846·43-s − 0.730·45-s + 1.07·47-s + 0.142·49-s + 0.599·51-s + 1.59·53-s + 0.435·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.799822888\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.799822888\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 + 0.0895T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 - 6.80T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 - 7.36T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 + 3.09T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 0.708T + 83T^{2} \) |
| 89 | \( 1 - 3.57T + 89T^{2} \) |
| 97 | \( 1 - 5.94T + 97T^{2} \) |
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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
−8.763617931000453299653413815200, −7.63245002920198582941213402704, −6.74683560228476523622410424962, −5.95268321548679877503409039465, −5.81021703035937947525350415293, −4.78555726048520835148049711881, −4.09957770392710766496532032749, −2.55564536526413268330600091848, −2.11281512662127715082794187653, −0.789509665780532702853107503053,
0.789509665780532702853107503053, 2.11281512662127715082794187653, 2.55564536526413268330600091848, 4.09957770392710766496532032749, 4.78555726048520835148049711881, 5.81021703035937947525350415293, 5.95268321548679877503409039465, 6.74683560228476523622410424962, 7.63245002920198582941213402704, 8.763617931000453299653413815200