L(s) = 1 | + 0.660·3-s + 2.45·5-s + 3.10·7-s − 2.56·9-s − 3.33·11-s − 5.08·13-s + 1.62·15-s − 17-s − 5.38·19-s + 2.05·21-s − 3.60·23-s + 1.01·25-s − 3.67·27-s − 0.718·29-s + 8.53·31-s − 2.20·33-s + 7.61·35-s + 2.73·37-s − 3.36·39-s − 1.77·41-s − 10.0·43-s − 6.28·45-s − 9.54·47-s + 2.64·49-s − 0.660·51-s − 8.25·53-s − 8.19·55-s + ⋯ |
L(s) = 1 | + 0.381·3-s + 1.09·5-s + 1.17·7-s − 0.854·9-s − 1.00·11-s − 1.41·13-s + 0.418·15-s − 0.242·17-s − 1.23·19-s + 0.447·21-s − 0.751·23-s + 0.203·25-s − 0.707·27-s − 0.133·29-s + 1.53·31-s − 0.384·33-s + 1.28·35-s + 0.449·37-s − 0.538·39-s − 0.277·41-s − 1.52·43-s − 0.937·45-s − 1.39·47-s + 0.377·49-s − 0.0925·51-s − 1.13·53-s − 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 0.660T + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 3.60T + 23T^{2} \) |
| 29 | \( 1 + 0.718T + 29T^{2} \) |
| 31 | \( 1 - 8.53T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.54T + 47T^{2} \) |
| 53 | \( 1 + 8.25T + 53T^{2} \) |
| 61 | \( 1 - 0.304T + 61T^{2} \) |
| 67 | \( 1 - 2.16T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 9.25T + 83T^{2} \) |
| 89 | \( 1 - 1.00T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.146052139277932937565797111198, −7.60109268723597148743690458222, −6.46521077200514965499338727447, −5.85517818053169678821399159737, −4.92126011780636330332783584807, −4.65055118275027703779307100849, −3.12264288376511381627467511916, −2.27403976753859290886056420512, −1.86596972900636316965011824082, 0,
1.86596972900636316965011824082, 2.27403976753859290886056420512, 3.12264288376511381627467511916, 4.65055118275027703779307100849, 4.92126011780636330332783584807, 5.85517818053169678821399159737, 6.46521077200514965499338727447, 7.60109268723597148743690458222, 8.146052139277932937565797111198