L(s) = 1 | − 3-s + 2·7-s + 9-s + 4·11-s − 6·13-s − 6·17-s − 2·21-s + 4·23-s − 27-s + 4·29-s − 10·31-s − 4·33-s − 2·37-s + 6·39-s − 2·41-s + 8·43-s − 12·47-s − 3·49-s + 6·51-s + 12·53-s + 4·59-s + 2·61-s + 2·63-s + 4·67-s − 4·69-s + 4·71-s + 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.436·21-s + 0.834·23-s − 0.192·27-s + 0.742·29-s − 1.79·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s + 0.520·59-s + 0.256·61-s + 0.251·63-s + 0.488·67-s − 0.481·69-s + 0.474·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.517055573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517055573\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.53668207412011919659530243812, −6.84445230805606459667372944565, −6.58739690122532465366966434700, −5.45362848812274335026775129730, −4.94477611665424568435490277562, −4.38194054884316272425505911313, −3.58531873675065089773666376742, −2.38347834548705836981541812186, −1.75846579157964817591495016051, −0.60361417218253467851429519685,
0.60361417218253467851429519685, 1.75846579157964817591495016051, 2.38347834548705836981541812186, 3.58531873675065089773666376742, 4.38194054884316272425505911313, 4.94477611665424568435490277562, 5.45362848812274335026775129730, 6.58739690122532465366966434700, 6.84445230805606459667372944565, 7.53668207412011919659530243812