| L(s) = 1 | − 0.0441·3-s − 1.35·5-s − 0.783·7-s − 2.99·9-s − 5.10·11-s − 4.89·13-s + 0.0599·15-s + 0.564·17-s − 6.52·19-s + 0.0345·21-s + 3.68·23-s − 3.15·25-s + 0.264·27-s + 1.53·29-s − 3.82·31-s + 0.225·33-s + 1.06·35-s + 5.45·37-s + 0.215·39-s − 7.43·41-s − 5.59·43-s + 4.07·45-s + 6.94·47-s − 6.38·49-s − 0.0248·51-s − 12.7·53-s + 6.94·55-s + ⋯ |
| L(s) = 1 | − 0.0254·3-s − 0.607·5-s − 0.296·7-s − 0.999·9-s − 1.53·11-s − 1.35·13-s + 0.0154·15-s + 0.136·17-s − 1.49·19-s + 0.00753·21-s + 0.767·23-s − 0.630·25-s + 0.0509·27-s + 0.285·29-s − 0.687·31-s + 0.0392·33-s + 0.180·35-s + 0.897·37-s + 0.0345·39-s − 1.16·41-s − 0.852·43-s + 0.607·45-s + 1.01·47-s − 0.912·49-s − 0.00348·51-s − 1.75·53-s + 0.936·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2542638429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2542638429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0441T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 + 0.783T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 0.564T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 - 5.45T + 37T^{2} \) |
| 41 | \( 1 + 7.43T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 - 2.97T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 3.83T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
| 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
−8.143521841557789531544809097515, −7.46476637941008252940178456663, −6.73778775979321367267822816526, −5.88734422237618643370200464987, −5.12513341793362791473103274596, −4.61542278629500549659296478098, −3.51059919883219905513444632249, −2.76435489256215358044861634404, −2.11767171093642129601886647916, −0.24205774895516122938623924880,
0.24205774895516122938623924880, 2.11767171093642129601886647916, 2.76435489256215358044861634404, 3.51059919883219905513444632249, 4.61542278629500549659296478098, 5.12513341793362791473103274596, 5.88734422237618643370200464987, 6.73778775979321367267822816526, 7.46476637941008252940178456663, 8.143521841557789531544809097515
