| L(s) = 1 | + 0.453·3-s + 3.09·5-s − 2.79·9-s − 3.54·11-s − 6.65·13-s + 1.40·15-s + 17-s + 3.06·19-s − 2.80·23-s + 4.59·25-s − 2.62·27-s + 1.12·29-s + 8.64·31-s − 1.60·33-s − 3.44·37-s − 3.01·39-s + 4.53·41-s − 10.2·43-s − 8.65·45-s − 6.83·47-s + 0.453·51-s − 6.24·53-s − 10.9·55-s + 1.39·57-s − 4.06·59-s − 12.4·61-s − 20.6·65-s + ⋯ |
| L(s) = 1 | + 0.261·3-s + 1.38·5-s − 0.931·9-s − 1.06·11-s − 1.84·13-s + 0.362·15-s + 0.242·17-s + 0.704·19-s − 0.585·23-s + 0.918·25-s − 0.505·27-s + 0.209·29-s + 1.55·31-s − 0.279·33-s − 0.567·37-s − 0.483·39-s + 0.708·41-s − 1.56·43-s − 1.29·45-s − 0.997·47-s + 0.0634·51-s − 0.857·53-s − 1.47·55-s + 0.184·57-s − 0.528·59-s − 1.59·61-s − 2.55·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.453T + 3T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 + 8.30T + 83T^{2} \) |
| 89 | \( 1 + 2.87T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−8.166126688699151470259588705867, −7.67406760130846390500338101990, −6.67457739130788157321885443614, −5.89037743951365806305072205064, −5.22618526258149314942714015160, −4.70284254052389108833019720146, −3.03377455998282125883053200618, −2.65188534621322103997445121611, −1.73083908675437238536008686165, 0,
1.73083908675437238536008686165, 2.65188534621322103997445121611, 3.03377455998282125883053200618, 4.70284254052389108833019720146, 5.22618526258149314942714015160, 5.89037743951365806305072205064, 6.67457739130788157321885443614, 7.67406760130846390500338101990, 8.166126688699151470259588705867
