| L(s) = 1 | − 3-s + 7-s + 9-s − 6.47·11-s − 4.47·13-s + 2·17-s + 2.47·19-s − 21-s + 4·23-s − 27-s − 2·29-s − 1.52·31-s + 6.47·33-s + 6.94·37-s + 4.47·39-s − 2·41-s + 8.94·43-s + 12.9·47-s + 49-s − 2·51-s + 3.52·53-s − 2.47·57-s + 8.94·59-s − 2·61-s + 63-s − 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s − 1.95·11-s − 1.24·13-s + 0.485·17-s + 0.567·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.371·29-s − 0.274·31-s + 1.12·33-s + 1.14·37-s + 0.716·39-s − 0.312·41-s + 1.36·43-s + 1.88·47-s + 0.142·49-s − 0.280·51-s + 0.484·53-s − 0.327·57-s + 1.16·59-s − 0.256·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−7.46038456531833449490161395909, −6.99656003258891582149796952613, −5.67478822161792585586748188426, −5.50205996174985757282984267805, −4.81529437929641885485740220246, −4.06582523867420475368323878026, −2.79837713643434620595470194847, −2.43024425338827948365226078628, −1.07915968727169713012636577153, 0,
1.07915968727169713012636577153, 2.43024425338827948365226078628, 2.79837713643434620595470194847, 4.06582523867420475368323878026, 4.81529437929641885485740220246, 5.50205996174985757282984267805, 5.67478822161792585586748188426, 6.99656003258891582149796952613, 7.46038456531833449490161395909
