| L(s) = 1 | − 1.15·3-s − 3.26·5-s − 1.92·7-s − 1.66·9-s + 1.97·11-s − 0.983·13-s + 3.76·15-s − 4.76·17-s + 4.17·19-s + 2.22·21-s + 8.45·23-s + 5.63·25-s + 5.39·27-s − 0.913·29-s + 7.19·31-s − 2.27·33-s + 6.29·35-s + 11.2·37-s + 1.13·39-s + 5.68·41-s − 9.63·43-s + 5.42·45-s − 4.21·47-s − 3.27·49-s + 5.50·51-s − 1.59·53-s − 6.42·55-s + ⋯ |
| L(s) = 1 | − 0.667·3-s − 1.45·5-s − 0.729·7-s − 0.554·9-s + 0.594·11-s − 0.272·13-s + 0.972·15-s − 1.15·17-s + 0.956·19-s + 0.486·21-s + 1.76·23-s + 1.12·25-s + 1.03·27-s − 0.169·29-s + 1.29·31-s − 0.396·33-s + 1.06·35-s + 1.84·37-s + 0.182·39-s + 0.887·41-s − 1.47·43-s + 0.809·45-s − 0.614·47-s − 0.467·49-s + 0.770·51-s − 0.219·53-s − 0.866·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 11 | \( 1 - 1.97T + 11T^{2} \) |
| 13 | \( 1 + 0.983T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 29 | \( 1 + 0.913T + 29T^{2} \) |
| 31 | \( 1 - 7.19T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 5.68T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 - 0.532T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 - 0.00198T + 89T^{2} \) |
| 97 | \( 1 - 4.11T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.015355473444848128269326988047, −7.27435887764755134544738308347, −6.60908829247568291700349457853, −6.02291652355185210084109489640, −4.85348513014998053748289831627, −4.44555963649198913943489762539, −3.32273822896803893088679932665, −2.82406232641215599308656511364, −0.987422918563694656532284449201, 0,
0.987422918563694656532284449201, 2.82406232641215599308656511364, 3.32273822896803893088679932665, 4.44555963649198913943489762539, 4.85348513014998053748289831627, 6.02291652355185210084109489640, 6.60908829247568291700349457853, 7.27435887764755134544738308347, 8.015355473444848128269326988047
