| L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s + 11-s + 6.81·13-s − 15-s + 17-s + 2.81·19-s + 3·21-s + 4.81·23-s + 25-s − 27-s + 3.81·29-s + 3.81·31-s − 33-s − 3·35-s + 37-s − 6.81·39-s + 5.81·41-s − 5.81·43-s + 45-s + 8·47-s + 2·49-s − 51-s − 10.6·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 0.333·9-s + 0.301·11-s + 1.89·13-s − 0.258·15-s + 0.242·17-s + 0.645·19-s + 0.654·21-s + 1.00·23-s + 0.200·25-s − 0.192·27-s + 0.708·29-s + 0.685·31-s − 0.174·33-s − 0.507·35-s + 0.164·37-s − 1.09·39-s + 0.908·41-s − 0.886·43-s + 0.149·45-s + 1.16·47-s + 0.285·49-s − 0.140·51-s − 1.46·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.989858825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989858825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 6.81T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 + 5.81T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 9.63T + 71T^{2} \) |
| 73 | \( 1 - 4.81T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6.81T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−7.64491677392019565186438955509, −6.80055475533782175769256024819, −6.30080634447999940739063982768, −5.91248185565661726184860969660, −5.10179688166973824217326357499, −4.20768055756261229443166045057, −3.40221105503283409814482163465, −2.83628616064925830508267680896, −1.46229804005906771736787856826, −0.77742377958555565099044729601,
0.77742377958555565099044729601, 1.46229804005906771736787856826, 2.83628616064925830508267680896, 3.40221105503283409814482163465, 4.20768055756261229443166045057, 5.10179688166973824217326357499, 5.91248185565661726184860969660, 6.30080634447999940739063982768, 6.80055475533782175769256024819, 7.64491677392019565186438955509
