L(s) = 1 | − 2-s + 1.78i·3-s + 4-s + (2.21 − 0.288i)5-s − 1.78i·6-s + 3.14i·7-s − 8-s − 0.191·9-s + (−2.21 + 0.288i)10-s − 0.908·11-s + 1.78i·12-s + 2.22·13-s − 3.14i·14-s + (0.515 + 3.96i)15-s + 16-s + 2.10·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.03i·3-s + 0.5·4-s + (0.991 − 0.129i)5-s − 0.729i·6-s + 1.19i·7-s − 0.353·8-s − 0.0638·9-s + (−0.701 + 0.0912i)10-s − 0.274·11-s + 0.515i·12-s + 0.616·13-s − 0.841i·14-s + (0.133 + 1.02i)15-s + 0.250·16-s + 0.509·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0524 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0524 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.848063 + 0.804688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.848063 + 0.804688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (-2.21 + 0.288i)T \) |
| 37 | \( 1 + (1.10 - 5.98i)T \) |
good | 3 | \( 1 - 1.78iT - 3T^{2} \) |
| 7 | \( 1 - 3.14iT - 7T^{2} \) |
| 11 | \( 1 + 0.908T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 4.16iT - 19T^{2} \) |
| 23 | \( 1 + 7.66T + 23T^{2} \) |
| 29 | \( 1 - 2.69iT - 29T^{2} \) |
| 31 | \( 1 - 5.96iT - 31T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 + 5.72T + 43T^{2} \) |
| 47 | \( 1 + 8.89iT - 47T^{2} \) |
| 53 | \( 1 + 9.37iT - 53T^{2} \) |
| 59 | \( 1 + 5.55iT - 59T^{2} \) |
| 61 | \( 1 - 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 7.64iT - 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 2.14iT - 73T^{2} \) |
| 79 | \( 1 + 3.35iT - 79T^{2} \) |
| 83 | \( 1 + 16.2iT - 83T^{2} \) |
| 89 | \( 1 + 8.35iT - 89T^{2} \) |
| 97 | \( 1 + 5.14T + 97T^{2} \) |
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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
−11.37497910416606447769736510039, −10.29952972534424313571894397360, −9.873319237429144766601952566215, −8.954913639755865302826710882537, −8.390049586890769144566011106766, −6.78747363313445153171335262825, −5.73501832478646686255905080357, −4.93214466796485927075300162484, −3.25735311500383785121737984036, −1.90476899800056057084679979235,
1.09669338376445786986416008131, 2.18381892287677649437099199069, 3.93717686973128500184503330128, 5.83079593375521467014380628849, 6.49842131276060442174703403325, 7.57977442408283067483058896221, 8.058520561651285588831565421431, 9.528979062470287457169428344913, 10.18706826507796261457118577573, 10.93311478416110175623544204684