L(s) = 1 | + 1.53·5-s + (0.414 − 2.61i)7-s − 0.585i·11-s − 2.16i·13-s − 5.86·17-s + 5.22i·19-s − 2.24i·23-s − 2.65·25-s − 5.41i·29-s − 4.32i·31-s + (0.634 − 4i)35-s − 4·37-s − 8.92·41-s + 10.4·43-s − 7.39·47-s + ⋯ |
L(s) = 1 | + 0.684·5-s + (0.156 − 0.987i)7-s − 0.176i·11-s − 0.600i·13-s − 1.42·17-s + 1.19i·19-s − 0.467i·23-s − 0.531·25-s − 1.00i·29-s − 0.777i·31-s + (0.107 − 0.676i)35-s − 0.657·37-s − 1.39·41-s + 1.59·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9406728206\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9406728206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.414 + 2.61i)T \) |
good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 11 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 + 2.16iT - 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 5.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.24iT - 23T^{2} \) |
| 29 | \( 1 + 5.41iT - 29T^{2} \) |
| 31 | \( 1 + 4.32iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.86T + 89T^{2} \) |
| 97 | \( 1 + 8.28iT - 97T^{2} \) |
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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
−7.987244368847751400838657639522, −7.55281755259654914800948751998, −6.46261923295529318220945724710, −6.10262412426113949287349596687, −5.11715658061707479696055530822, −4.28411211896231794266818762189, −3.58671610486253213078817872953, −2.41898796968907885261094171356, −1.56284705793649775867005365036, −0.24264439834399897995944643148,
1.62488931471116876300727004175, 2.27504958246501217144529474002, 3.17115358202344242756802451574, 4.37737003480721652731239308888, 5.08649696671687619751010475374, 5.72683666148033295982296507988, 6.69727978264729022787346293089, 6.99276192483430576864900799241, 8.237692818934471331438006356687, 8.891751201839696278566610305225