L(s) = 1 | + 2.39·3-s − 3.16·5-s + 3.22·7-s + 2.71·9-s + 3.58·11-s − 7.07·13-s − 7.56·15-s + 6.97·17-s + 5.21·19-s + 7.71·21-s + 3.09·23-s + 5.01·25-s − 0.685·27-s − 3.23·29-s + 3.98·31-s + 8.57·33-s − 10.2·35-s + 7.10·37-s − 16.8·39-s − 2.86·41-s − 9.31·43-s − 8.58·45-s + 11.6·47-s + 3.42·49-s + 16.6·51-s − 11.3·53-s − 11.3·55-s + ⋯ |
L(s) = 1 | + 1.37·3-s − 1.41·5-s + 1.22·7-s + 0.904·9-s + 1.08·11-s − 1.96·13-s − 1.95·15-s + 1.69·17-s + 1.19·19-s + 1.68·21-s + 0.645·23-s + 1.00·25-s − 0.132·27-s − 0.600·29-s + 0.716·31-s + 1.49·33-s − 1.72·35-s + 1.16·37-s − 2.70·39-s − 0.446·41-s − 1.42·43-s − 1.28·45-s + 1.69·47-s + 0.489·49-s + 2.33·51-s − 1.55·53-s − 1.53·55-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
\(L(1)\) |
\(\approx\) |
\(3.314460322\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.314460322\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 - 7.10T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 3.17T + 73T^{2} \) |
| 79 | \( 1 + 3.67T + 79T^{2} \) |
| 83 | \( 1 - 0.597T + 83T^{2} \) |
| 89 | \( 1 - 3.80T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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
−7.88015336268025562699142289267, −7.48223992553460977017843796213, −6.92991019811446136707806458996, −5.50072902585020088283833187392, −4.80840327454359911572212742759, −4.19030682625203021771616382290, −3.38831719598242828984270549094, −2.90731803721535929264599761336, −1.83952652101507794238812151552, −0.877725039519148317495077592262,
0.877725039519148317495077592262, 1.83952652101507794238812151552, 2.90731803721535929264599761336, 3.38831719598242828984270549094, 4.19030682625203021771616382290, 4.80840327454359911572212742759, 5.50072902585020088283833187392, 6.92991019811446136707806458996, 7.48223992553460977017843796213, 7.88015336268025562699142289267