L(s) = 1 | − 2.55·2-s + 4.51·4-s − 3.42·5-s + 7-s − 6.41·8-s + 8.75·10-s − 2.17·13-s − 2.55·14-s + 7.33·16-s + 4.51·17-s − 2.42·19-s − 15.4·20-s − 0.648·23-s + 6.75·25-s + 5.55·26-s + 4.51·28-s + 1.25·29-s − 8.03·31-s − 5.90·32-s − 11.5·34-s − 3.42·35-s + 5.01·37-s + 6.18·38-s + 21.9·40-s + 2.62·41-s − 1.46·43-s + 1.65·46-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 2.25·4-s − 1.53·5-s + 0.377·7-s − 2.26·8-s + 2.76·10-s − 0.603·13-s − 0.682·14-s + 1.83·16-s + 1.09·17-s − 0.556·19-s − 3.45·20-s − 0.135·23-s + 1.35·25-s + 1.08·26-s + 0.852·28-s + 0.232·29-s − 1.44·31-s − 1.04·32-s − 1.97·34-s − 0.579·35-s + 0.824·37-s + 1.00·38-s + 3.47·40-s + 0.409·41-s − 0.222·43-s + 0.243·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 0.648T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 - 5.04T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 7.66T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 6.22T + 67T^{2} \) |
| 71 | \( 1 + 4.22T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 9.76T + 79T^{2} \) |
| 83 | \( 1 - 8.37T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−7.79607180428168816343905489156, −7.26846772917002783082955672082, −6.58167633854256035084133550522, −5.61110431322419641011122381689, −4.61395307873377013212955991821, −3.74496763312182612608797484862, −2.91002804611702792917356377945, −1.93544757907957635512886099955, −0.896429490626231175202944561293, 0,
0.896429490626231175202944561293, 1.93544757907957635512886099955, 2.91002804611702792917356377945, 3.74496763312182612608797484862, 4.61395307873377013212955991821, 5.61110431322419641011122381689, 6.58167633854256035084133550522, 7.26846772917002783082955672082, 7.79607180428168816343905489156