L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 6·11-s + 6·13-s + 16-s − 4·19-s − 20-s − 6·22-s + 25-s + 6·26-s + 8·29-s + 2·31-s + 32-s + 4·37-s − 4·38-s − 40-s + 10·41-s − 6·43-s − 6·44-s + 2·47-s + 50-s + 6·52-s − 10·53-s + 6·55-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.80·11-s + 1.66·13-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s + 1.17·26-s + 1.48·29-s + 0.359·31-s + 0.176·32-s + 0.657·37-s − 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.914·43-s − 0.904·44-s + 0.291·47-s + 0.141·50-s + 0.832·52-s − 1.37·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.637476455\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.637476455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.160279983269438934100881939312, −7.79170468641312060938137918332, −6.68867036961452943989019141336, −6.16236649160420655702633062986, −5.33348176211912876532962783406, −4.61091460088952365750923964785, −3.84585699504002316995217387933, −2.99348228626267769382056994446, −2.22774537907891883805264475213, −0.814028269383723992773983515515,
0.814028269383723992773983515515, 2.22774537907891883805264475213, 2.99348228626267769382056994446, 3.84585699504002316995217387933, 4.61091460088952365750923964785, 5.33348176211912876532962783406, 6.16236649160420655702633062986, 6.68867036961452943989019141336, 7.79170468641312060938137918332, 8.160279983269438934100881939312