| L(s) = 1 | + 2.98·3-s − 3.49·5-s − 1.06·7-s + 5.88·9-s − 11-s − 0.0563·13-s − 10.4·15-s − 4.53·17-s + 19-s − 3.18·21-s + 1.07·23-s + 7.19·25-s + 8.59·27-s + 0.299·29-s − 9.18·31-s − 2.98·33-s + 3.72·35-s + 4.50·37-s − 0.167·39-s + 12.0·41-s − 10.7·43-s − 20.5·45-s − 2.89·47-s − 5.86·49-s − 13.5·51-s − 12.3·53-s + 3.49·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s − 1.56·5-s − 0.403·7-s + 1.96·9-s − 0.301·11-s − 0.0156·13-s − 2.68·15-s − 1.10·17-s + 0.229·19-s − 0.694·21-s + 0.225·23-s + 1.43·25-s + 1.65·27-s + 0.0556·29-s − 1.64·31-s − 0.518·33-s + 0.630·35-s + 0.740·37-s − 0.0268·39-s + 1.87·41-s − 1.63·43-s − 3.06·45-s − 0.422·47-s − 0.837·49-s − 1.89·51-s − 1.69·53-s + 0.470·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 13 | \( 1 + 0.0563T + 13T^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 0.299T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 - 4.50T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 8.09T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 0.183T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−8.214803737777390179134356684742, −7.57932761179425134646029797965, −7.25298759937769780827013989211, −6.22307377998990891525157536429, −4.72349457424629295371279946153, −4.18162946087953513154160259659, −3.33450498956722219457749929192, −2.89136108782975314762520037784, −1.71300969595905993635703838786, 0,
1.71300969595905993635703838786, 2.89136108782975314762520037784, 3.33450498956722219457749929192, 4.18162946087953513154160259659, 4.72349457424629295371279946153, 6.22307377998990891525157536429, 7.25298759937769780827013989211, 7.57932761179425134646029797965, 8.214803737777390179134356684742
