| L(s) = 1 | − 1.38·3-s − 1.38·5-s + 2.44·7-s − 1.07·9-s − 1.05·11-s + 13-s + 1.92·15-s + 6.96·17-s − 0.408·19-s − 3.38·21-s − 3.57·23-s − 3.07·25-s + 5.65·27-s − 4.34·29-s + 5.74·31-s + 1.46·33-s − 3.38·35-s − 7.49·37-s − 1.38·39-s + 1.57·41-s − 2.27·43-s + 1.49·45-s + 4.33·47-s − 1.03·49-s − 9.65·51-s + 0.647·53-s + 1.46·55-s + ⋯ |
| L(s) = 1 | − 0.800·3-s − 0.620·5-s + 0.923·7-s − 0.359·9-s − 0.318·11-s + 0.277·13-s + 0.496·15-s + 1.68·17-s − 0.0936·19-s − 0.738·21-s − 0.745·23-s − 0.615·25-s + 1.08·27-s − 0.807·29-s + 1.03·31-s + 0.254·33-s − 0.572·35-s − 1.23·37-s − 0.222·39-s + 0.246·41-s − 0.346·43-s + 0.222·45-s + 0.631·47-s − 0.147·49-s − 1.35·51-s + 0.0889·53-s + 0.197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131546512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131546512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 + 0.408T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 - 0.647T + 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 + 7.68T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−8.291494927874535444578633804908, −8.013885567516083623122765434841, −7.24641385344615540209820055135, −6.20419729341872332964163797542, −5.53634501552543569702746194404, −4.97855554165716782326272172055, −4.00875371243866630936741782841, −3.17254465117049566015418313949, −1.87163539541074727402423695113, −0.66521293326918350048781968775,
0.66521293326918350048781968775, 1.87163539541074727402423695113, 3.17254465117049566015418313949, 4.00875371243866630936741782841, 4.97855554165716782326272172055, 5.53634501552543569702746194404, 6.20419729341872332964163797542, 7.24641385344615540209820055135, 8.013885567516083623122765434841, 8.291494927874535444578633804908
