L(s) = 1 | + 2.48·3-s + 1.56·5-s − 7-s + 3.15·9-s + 11-s + 13-s + 3.87·15-s + 1.17·17-s + 6.31·19-s − 2.48·21-s + 4.27·23-s − 2.56·25-s + 0.390·27-s − 6.71·29-s − 3.78·31-s + 2.48·33-s − 1.56·35-s + 2.14·37-s + 2.48·39-s + 11.7·41-s − 4.31·43-s + 4.92·45-s − 3.97·47-s + 49-s + 2.92·51-s + 4.15·53-s + 1.56·55-s + ⋯ |
L(s) = 1 | + 1.43·3-s + 0.698·5-s − 0.377·7-s + 1.05·9-s + 0.301·11-s + 0.277·13-s + 1.00·15-s + 0.285·17-s + 1.44·19-s − 0.541·21-s + 0.890·23-s − 0.512·25-s + 0.0752·27-s − 1.24·29-s − 0.680·31-s + 0.431·33-s − 0.263·35-s + 0.352·37-s + 0.397·39-s + 1.83·41-s − 0.657·43-s + 0.734·45-s − 0.579·47-s + 0.142·49-s + 0.409·51-s + 0.570·53-s + 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.366352526\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.366352526\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 23 | \( 1 - 4.27T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 4.31T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 0.0156T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 - 5.87T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.76528006942770604144727041250, −7.38151697167181273736637018800, −6.55805145303561023215797792282, −5.69475746344510435471859661051, −5.14826439902232917326422020201, −3.86574229555183158164542493490, −3.52345351135343307666220430315, −2.65598711605861066583276747183, −1.98642950358766981309839414651, −1.00685727346929222851253481678,
1.00685727346929222851253481678, 1.98642950358766981309839414651, 2.65598711605861066583276747183, 3.52345351135343307666220430315, 3.86574229555183158164542493490, 5.14826439902232917326422020201, 5.69475746344510435471859661051, 6.55805145303561023215797792282, 7.38151697167181273736637018800, 7.76528006942770604144727041250