| L(s) = 1 | − 2.33·2-s + 3.44·4-s + 0.335·5-s − 7-s − 3.37·8-s − 0.783·10-s + 5.16·11-s − 5.99·13-s + 2.33·14-s + 0.987·16-s − 7.35·17-s + 5.24·19-s + 1.15·20-s − 12.0·22-s − 1.66·23-s − 4.88·25-s + 14.0·26-s − 3.44·28-s − 1.61·29-s − 6.38·31-s + 4.44·32-s + 17.1·34-s − 0.335·35-s − 11.1·37-s − 12.2·38-s − 1.13·40-s + 3.12·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.72·4-s + 0.150·5-s − 0.377·7-s − 1.19·8-s − 0.247·10-s + 1.55·11-s − 1.66·13-s + 0.623·14-s + 0.246·16-s − 1.78·17-s + 1.20·19-s + 0.258·20-s − 2.57·22-s − 0.347·23-s − 0.977·25-s + 2.74·26-s − 0.651·28-s − 0.299·29-s − 1.14·31-s + 0.786·32-s + 2.94·34-s − 0.0567·35-s − 1.82·37-s − 1.98·38-s − 0.179·40-s + 0.488·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4578289032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4578289032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 - 0.335T + 5T^{2} \) |
| 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + 5.99T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 8.78T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 - 0.675T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.75035962947698462599105625015, −7.30139791728170838513075751921, −6.76591803144007963826073577019, −6.14306477711785679436007704000, −5.10806442763056051850461105657, −4.22634995900053371680837875586, −3.30053479247226116330307339374, −2.15705616730913714315960910888, −1.73028949378240316132940158945, −0.41945031381381770228230049071,
0.41945031381381770228230049071, 1.73028949378240316132940158945, 2.15705616730913714315960910888, 3.30053479247226116330307339374, 4.22634995900053371680837875586, 5.10806442763056051850461105657, 6.14306477711785679436007704000, 6.76591803144007963826073577019, 7.30139791728170838513075751921, 7.75035962947698462599105625015
