| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s − 3·8-s + 9-s + 10-s − 6·11-s + 12-s − 3·13-s − 15-s − 16-s + 6·17-s + 18-s − 6·19-s − 20-s − 6·22-s − 9·23-s + 3·24-s − 4·25-s − 3·26-s − 27-s + 5·29-s − 30-s + 8·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s + 0.288·12-s − 0.832·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 1.27·22-s − 1.87·23-s + 0.612·24-s − 4/5·25-s − 0.588·26-s − 0.192·27-s + 0.928·29-s − 0.182·30-s + 1.43·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9584157675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9584157675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−8.142248505391969044333522137528, −7.35208926553093138358146858520, −6.29294979913722372770234361732, −5.77586232383079649098162060526, −5.22686740337808770106592607958, −4.61790864994148116621323147930, −3.84124889874107073459981589057, −2.80511701413050653722753158758, −2.08738541741165147720171543934, −0.45035237422559106042311154130,
0.45035237422559106042311154130, 2.08738541741165147720171543934, 2.80511701413050653722753158758, 3.84124889874107073459981589057, 4.61790864994148116621323147930, 5.22686740337808770106592607958, 5.77586232383079649098162060526, 6.29294979913722372770234361732, 7.35208926553093138358146858520, 8.142248505391969044333522137528
