L(s) = 1 | + 0.302·3-s − 1.30·5-s + 4.60·7-s − 2.90·9-s − 1.30·11-s + 2.30·13-s − 0.394·15-s − 6·17-s − 2·19-s + 1.39·21-s − 6.90·23-s − 3.30·25-s − 1.78·27-s − 6.90·29-s + 3.30·31-s − 0.394·33-s − 6·35-s − 37-s + 0.697·39-s − 0.908·41-s + 6.60·43-s + 3.78·45-s − 2.60·47-s + 14.2·49-s − 1.81·51-s + 6·53-s + 1.69·55-s + ⋯ |
L(s) = 1 | + 0.174·3-s − 0.582·5-s + 1.74·7-s − 0.969·9-s − 0.392·11-s + 0.638·13-s − 0.101·15-s − 1.45·17-s − 0.458·19-s + 0.304·21-s − 1.44·23-s − 0.660·25-s − 0.344·27-s − 1.28·29-s + 0.593·31-s − 0.0686·33-s − 1.01·35-s − 0.164·37-s + 0.111·39-s − 0.141·41-s + 1.00·43-s + 0.564·45-s − 0.380·47-s + 2.03·49-s − 0.254·51-s + 0.824·53-s + 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 0.302T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 41 | \( 1 + 0.908T + 41T^{2} \) |
| 43 | \( 1 - 6.60T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.21T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.488195384857996670944557521108, −7.985610171977074464649569877318, −7.32088028196587282419508517849, −6.10223774884279333131430876112, −5.46046707529506985651634671878, −4.41933973972685060954460496120, −3.92677720540305487400794133529, −2.52088968681583352688459548288, −1.73378050295571585239151369541, 0,
1.73378050295571585239151369541, 2.52088968681583352688459548288, 3.92677720540305487400794133529, 4.41933973972685060954460496120, 5.46046707529506985651634671878, 6.10223774884279333131430876112, 7.32088028196587282419508517849, 7.985610171977074464649569877318, 8.488195384857996670944557521108