| L(s) = 1 | − 1.49·3-s − 1.09·5-s + 0.799·7-s − 0.776·9-s + 0.838·11-s + 2.03·13-s + 1.63·15-s − 1.82·17-s − 4.30·19-s − 1.19·21-s + 0.747·23-s − 3.79·25-s + 5.63·27-s + 2.79·29-s + 0.0226·31-s − 1.25·33-s − 0.877·35-s + 1.80·37-s − 3.03·39-s + 6.90·41-s + 8.84·43-s + 0.852·45-s + 6.50·47-s − 6.36·49-s + 2.71·51-s − 8.42·53-s − 0.920·55-s + ⋯ |
| L(s) = 1 | − 0.860·3-s − 0.490·5-s + 0.302·7-s − 0.258·9-s + 0.252·11-s + 0.563·13-s + 0.422·15-s − 0.442·17-s − 0.987·19-s − 0.260·21-s + 0.155·23-s − 0.759·25-s + 1.08·27-s + 0.518·29-s + 0.00406·31-s − 0.217·33-s − 0.148·35-s + 0.296·37-s − 0.485·39-s + 1.07·41-s + 1.34·43-s + 0.127·45-s + 0.948·47-s − 0.908·49-s + 0.380·51-s − 1.15·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 7 | \( 1 - 0.799T + 7T^{2} \) |
| 11 | \( 1 - 0.838T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 - 0.747T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 - 0.0226T + 31T^{2} \) |
| 37 | \( 1 - 1.80T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 - 6.50T + 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 - 5.12T + 83T^{2} \) |
| 89 | \( 1 + 9.57T + 89T^{2} \) |
| 97 | \( 1 + 0.843T + 97T^{2} \) |
| 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
−7.55095820436264941692082324640, −6.55543918451921476210163395361, −6.16058186904638036169520008670, −5.50318006436770672153346388434, −4.55231711705508960618467916077, −4.16690850705804611557406736346, −3.14016788406647018159486109057, −2.18536983472237648809626744480, −1.04124368194136027110486166907, 0,
1.04124368194136027110486166907, 2.18536983472237648809626744480, 3.14016788406647018159486109057, 4.16690850705804611557406736346, 4.55231711705508960618467916077, 5.50318006436770672153346388434, 6.16058186904638036169520008670, 6.55543918451921476210163395361, 7.55095820436264941692082324640
