L(s) = 1 | − 2.63·2-s − 0.0191·3-s + 4.95·4-s − 5-s + 0.0505·6-s + 0.546·7-s − 7.77·8-s − 2.99·9-s + 2.63·10-s + 11-s − 0.0949·12-s − 3.82·13-s − 1.44·14-s + 0.0191·15-s + 10.6·16-s − 5.14·17-s + 7.90·18-s + 6.92·19-s − 4.95·20-s − 0.0104·21-s − 2.63·22-s − 3.56·23-s + 0.149·24-s + 25-s + 10.0·26-s + 0.115·27-s + 2.70·28-s + ⋯ |
L(s) = 1 | − 1.86·2-s − 0.0110·3-s + 2.47·4-s − 0.447·5-s + 0.0206·6-s + 0.206·7-s − 2.74·8-s − 0.999·9-s + 0.833·10-s + 0.301·11-s − 0.0274·12-s − 1.06·13-s − 0.385·14-s + 0.00495·15-s + 2.65·16-s − 1.24·17-s + 1.86·18-s + 1.58·19-s − 1.10·20-s − 0.00228·21-s − 0.562·22-s − 0.744·23-s + 0.0304·24-s + 0.200·25-s + 1.97·26-s + 0.0221·27-s + 0.511·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 0.0191T + 3T^{2} \) |
| 7 | \( 1 - 0.546T + 7T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 - 0.258T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 + 5.33T + 61T^{2} \) |
| 67 | \( 1 - 7.19T + 67T^{2} \) |
| 71 | \( 1 - 7.72T + 71T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 8.13T + 89T^{2} \) |
| 97 | \( 1 + 6.00T + 97T^{2} \) |
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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
−8.067225072742153069277565838054, −7.67550796680082776317940727712, −6.93855510350576982785940105539, −6.19810031919569712126955718761, −5.32448514685641035511529844862, −4.15817080300950388865401163186, −2.83888238085385407405117169610, −2.35302534356720801849157631428, −1.03637790771142642909599607390, 0,
1.03637790771142642909599607390, 2.35302534356720801849157631428, 2.83888238085385407405117169610, 4.15817080300950388865401163186, 5.32448514685641035511529844862, 6.19810031919569712126955718761, 6.93855510350576982785940105539, 7.67550796680082776317940727712, 8.067225072742153069277565838054