L(s) = 1 | + 2-s − 4-s + 5-s + 2·7-s − 3·8-s + 10-s + 4·11-s − 13-s + 2·14-s − 16-s + 4·17-s + 6·19-s − 20-s + 4·22-s + 25-s − 26-s − 2·28-s + 4·29-s − 10·31-s + 5·32-s + 4·34-s + 2·35-s − 2·37-s + 6·38-s − 3·40-s + 6·41-s − 8·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s − 1.06·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s + 0.970·17-s + 1.37·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.196·26-s − 0.377·28-s + 0.742·29-s − 1.79·31-s + 0.883·32-s + 0.685·34-s + 0.338·35-s − 0.328·37-s + 0.973·38-s − 0.474·40-s + 0.937·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.136169645\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.136169645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.78754913503442851991561141736, −9.606143923072779152442392232211, −9.158186635818496855923068127670, −8.069603374513132812350986247673, −7.01279831140022405844648391421, −5.80203985872581797785224323059, −5.17362861953367702283649965368, −4.13601102912022186614527931542, −3.11999237486850599731326861173, −1.37276923078037774007010116691,
1.37276923078037774007010116691, 3.11999237486850599731326861173, 4.13601102912022186614527931542, 5.17362861953367702283649965368, 5.80203985872581797785224323059, 7.01279831140022405844648391421, 8.069603374513132812350986247673, 9.158186635818496855923068127670, 9.606143923072779152442392232211, 10.78754913503442851991561141736