L(s) = 1 | − 1.93i·2-s − 2.54·3-s − 1.74·4-s − 0.312i·5-s + 4.92i·6-s − i·7-s − 0.495i·8-s + 3.48·9-s − 0.603·10-s + 4.16i·11-s + 4.43·12-s − 1.93·14-s + 0.794i·15-s − 4.44·16-s + 5.20·17-s − 6.73i·18-s + ⋯ |
L(s) = 1 | − 1.36i·2-s − 1.46·3-s − 0.871·4-s − 0.139i·5-s + 2.01i·6-s − 0.377i·7-s − 0.175i·8-s + 1.16·9-s − 0.190·10-s + 1.25i·11-s + 1.28·12-s − 0.517·14-s + 0.205i·15-s − 1.11·16-s + 1.26·17-s − 1.58i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9257927855\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9257927855\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.93iT - 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 0.312iT - 5T^{2} \) |
| 11 | \( 1 - 4.16iT - 11T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 - 4.87iT - 19T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 9.52iT - 31T^{2} \) |
| 37 | \( 1 + 11.7iT - 37T^{2} \) |
| 41 | \( 1 - 0.433iT - 41T^{2} \) |
| 43 | \( 1 - 8.96T + 43T^{2} \) |
| 47 | \( 1 - 8.62iT - 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + 11.7iT - 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 8.71iT - 67T^{2} \) |
| 71 | \( 1 + 6.09iT - 71T^{2} \) |
| 73 | \( 1 + 3.59iT - 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 - 0.500iT - 83T^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 - 2.35iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.969531702849790185144800176701, −9.279363190392574526272463153878, −7.83564872669325909816281979445, −6.96730793526039337244710727710, −6.08710846472854142314662871060, −5.06395344290899323628732408148, −4.34539417571230829540726425970, −3.31686402516441776766854912673, −1.83921944994878094277127535548, −0.860582900003912876305567097968,
0.72632906423735141949661504142, 2.83903052340138230685223863982, 4.43042592150593150938547526846, 5.33657863216841373739710636188, 5.86360740805907588753224644611, 6.41576880029613254467008494159, 7.23139907203513451870325329842, 8.152738865665400423062667469946, 8.866935750636166803855512619921, 9.997439478968086186045780671362