| L(s) = 1 | + 2.27·3-s − 1.12·5-s − 2.61·7-s + 2.18·9-s − 2.61·11-s − 2.55·15-s − 5.55·17-s − 7.94·19-s − 5.94·21-s − 2.27·23-s − 3.74·25-s − 1.85·27-s + 7.79·29-s + 8.26·31-s − 5.94·33-s + 2.92·35-s − 3.60·37-s + 5.73·41-s + 4.03·43-s − 2.45·45-s − 2.66·47-s − 0.186·49-s − 12.6·51-s − 9.54·53-s + 2.92·55-s − 18.0·57-s + 3.83·59-s + ⋯ |
| L(s) = 1 | + 1.31·3-s − 0.501·5-s − 0.986·7-s + 0.728·9-s − 0.787·11-s − 0.659·15-s − 1.34·17-s − 1.82·19-s − 1.29·21-s − 0.474·23-s − 0.748·25-s − 0.356·27-s + 1.44·29-s + 1.48·31-s − 1.03·33-s + 0.494·35-s − 0.592·37-s + 0.895·41-s + 0.615·43-s − 0.365·45-s − 0.388·47-s − 0.0266·49-s − 1.77·51-s − 1.31·53-s + 0.394·55-s − 2.39·57-s + 0.498·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 17 | \( 1 + 5.55T + 17T^{2} \) |
| 19 | \( 1 + 7.94T + 19T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 + 1.76T + 67T^{2} \) |
| 71 | \( 1 + 1.01T + 71T^{2} \) |
| 73 | \( 1 + 0.323T + 73T^{2} \) |
| 79 | \( 1 + 2.66T + 79T^{2} \) |
| 83 | \( 1 + 2.77T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−9.032117498018715549465155519473, −8.367795419941493467982947458937, −7.895849939927543009048780307710, −6.76213997010222392532564429924, −6.15422154653012693751085479406, −4.59779173330925976419146230784, −3.90503004272240074557166839904, −2.84471593770318898754249638357, −2.23471177482404141069780499187, 0,
2.23471177482404141069780499187, 2.84471593770318898754249638357, 3.90503004272240074557166839904, 4.59779173330925976419146230784, 6.15422154653012693751085479406, 6.76213997010222392532564429924, 7.895849939927543009048780307710, 8.367795419941493467982947458937, 9.032117498018715549465155519473
