| L(s) = 1 | − 2·4-s − 2.44·5-s − 4.44·7-s + 13-s + 4·16-s + 1.44·17-s − 19-s + 4.89·20-s + 23-s + 0.999·25-s + 8.89·28-s − 29-s − 10.4·31-s + 10.8·35-s + 9.89·37-s + 9.34·41-s − 0.101·43-s + 12.4·47-s + 12.7·49-s − 2·52-s + 2·53-s + 0.550·59-s − 3.10·61-s − 8·64-s − 2.44·65-s − 8.89·67-s − 2.89·68-s + ⋯ |
| L(s) = 1 | − 4-s − 1.09·5-s − 1.68·7-s + 0.277·13-s + 16-s + 0.351·17-s − 0.229·19-s + 1.09·20-s + 0.208·23-s + 0.199·25-s + 1.68·28-s − 0.185·29-s − 1.87·31-s + 1.84·35-s + 1.62·37-s + 1.45·41-s − 0.0154·43-s + 1.81·47-s + 1.82·49-s − 0.277·52-s + 0.274·53-s + 0.0716·59-s − 0.397·61-s − 64-s − 0.303·65-s − 1.08·67-s − 0.351·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 - 9.34T + 41T^{2} \) |
| 43 | \( 1 + 0.101T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 0.550T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 + 5.44T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.57382637482461060688925434005, −7.30725985450537149508723027077, −6.11849428646037861499601062937, −5.74802905672627806791710638245, −4.62696712134164529345462278958, −3.83251270554898348963212431355, −3.57824233798080791223310775096, −2.59424369458930025707687990159, −0.862305715804540496833711459923, 0,
0.862305715804540496833711459923, 2.59424369458930025707687990159, 3.57824233798080791223310775096, 3.83251270554898348963212431355, 4.62696712134164529345462278958, 5.74802905672627806791710638245, 6.11849428646037861499601062937, 7.30725985450537149508723027077, 7.57382637482461060688925434005
