L(s) = 1 | − 5-s + 11-s + 13-s + 7·17-s − 2·19-s − 3·23-s + 25-s − 6·29-s − 8·31-s − 37-s + 9·41-s + 2·43-s + 8·47-s + 6·53-s − 55-s + 3·59-s − 14·61-s − 65-s + 15·67-s + 8·71-s + 73-s − 79-s + 8·83-s − 7·85-s + 15·89-s + 2·95-s + 3·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s + 0.277·13-s + 1.69·17-s − 0.458·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.164·37-s + 1.40·41-s + 0.304·43-s + 1.16·47-s + 0.824·53-s − 0.134·55-s + 0.390·59-s − 1.79·61-s − 0.124·65-s + 1.83·67-s + 0.949·71-s + 0.117·73-s − 0.112·79-s + 0.878·83-s − 0.759·85-s + 1.58·89-s + 0.205·95-s + 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.12178982541576, −12.52654245958536, −12.31843029600230, −11.84911259672785, −11.30023323762013, −10.74880833108262, −10.58403540084691, −9.847101778464591, −9.326333073639923, −9.098489981120598, −8.405172523071667, −7.872003157237617, −7.524245098291879, −7.190685811872473, −6.341594059467672, −6.047712845544009, −5.341568263839371, −5.161027546736063, −4.171678296913767, −3.754619759603122, −3.596225104318705, −2.668404384566366, −2.159635281559504, −1.393304596802603, −0.8290221597073889, 0,
0.8290221597073889, 1.393304596802603, 2.159635281559504, 2.668404384566366, 3.596225104318705, 3.754619759603122, 4.171678296913767, 5.161027546736063, 5.341568263839371, 6.047712845544009, 6.341594059467672, 7.190685811872473, 7.524245098291879, 7.872003157237617, 8.405172523071667, 9.098489981120598, 9.326333073639923, 9.847101778464591, 10.58403540084691, 10.74880833108262, 11.30023323762013, 11.84911259672785, 12.31843029600230, 12.52654245958536, 13.12178982541576