| L(s) = 1 | − 1.95·2-s − 0.209·3-s + 2.82·4-s + 0.408·6-s − 1.61·7-s − 3.57·8-s − 0.956·9-s − 11-s − 0.591·12-s + 3.16·14-s + 4.16·16-s + 1.33·17-s + 1.87·18-s + 19-s + 0.338·21-s + 1.95·22-s + 0.747·24-s + 25-s + 0.408·27-s − 4.57·28-s + 1.82·31-s − 4.57·32-s + 0.209·33-s − 2.61·34-s − 2.70·36-s − 1.95·38-s + 0.618·41-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 0.209·3-s + 2.82·4-s + 0.408·6-s − 1.61·7-s − 3.57·8-s − 0.956·9-s − 11-s − 0.591·12-s + 3.16·14-s + 4.16·16-s + 1.33·17-s + 1.87·18-s + 19-s + 0.338·21-s + 1.95·22-s + 0.747·24-s + 25-s + 0.408·27-s − 4.57·28-s + 1.82·31-s − 4.57·32-s + 0.209·33-s − 2.61·34-s − 2.70·36-s − 1.95·38-s + 0.618·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2852444119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2852444119\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 2 | \( 1 + 1.95T + T^{2} \) |
| 3 | \( 1 + 0.209T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.33T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.82T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 + 0.209T + T^{2} \) |
| 47 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.20752108264363463514394172634, −9.368025793393212441114541410280, −8.649722360148298575808290556684, −7.80871489652062796102556172296, −7.07912208381095880146905738094, −6.16627689587595535822287577686, −5.54672924703800501276147448678, −3.07704057748810479417253793137, −2.79842013256068495669075642492, −0.794758626314541911384868175072,
0.794758626314541911384868175072, 2.79842013256068495669075642492, 3.07704057748810479417253793137, 5.54672924703800501276147448678, 6.16627689587595535822287577686, 7.07912208381095880146905738094, 7.80871489652062796102556172296, 8.649722360148298575808290556684, 9.368025793393212441114541410280, 10.20752108264363463514394172634
