| L(s) = 1 | − 7-s − 2·11-s + 2·13-s − 8·17-s + 2·19-s + 6·29-s − 6·31-s + 8·37-s − 6·41-s − 8·43-s + 4·47-s + 49-s + 2·53-s − 8·59-s + 10·61-s + 12·67-s − 14·71-s − 10·73-s + 2·77-s − 4·79-s + 16·83-s − 10·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.603·11-s + 0.554·13-s − 1.94·17-s + 0.458·19-s + 1.11·29-s − 1.07·31-s + 1.31·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.274·53-s − 1.04·59-s + 1.28·61-s + 1.46·67-s − 1.66·71-s − 1.17·73-s + 0.227·77-s − 0.450·79-s + 1.75·83-s − 1.05·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301919733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301919733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.45086834532493, −14.89131359692200, −14.30393597272562, −13.43463758355915, −13.38030396311992, −12.84469347318119, −12.10939073695306, −11.46815866508144, −11.10598169351534, −10.39080982111493, −10.03855461300171, −9.142516202635659, −8.871554131418826, −8.185182995107149, −7.599355324624315, −6.767358203556948, −6.528721355217858, −5.729298754511507, −5.079853041567526, −4.425811312623004, −3.795645846501249, −2.978283864835394, −2.384428617201535, −1.542417228614689, −0.4449827709280319,
0.4449827709280319, 1.542417228614689, 2.384428617201535, 2.978283864835394, 3.795645846501249, 4.425811312623004, 5.079853041567526, 5.729298754511507, 6.528721355217858, 6.767358203556948, 7.599355324624315, 8.185182995107149, 8.871554131418826, 9.142516202635659, 10.03855461300171, 10.39080982111493, 11.10598169351534, 11.46815866508144, 12.10939073695306, 12.84469347318119, 13.38030396311992, 13.43463758355915, 14.30393597272562, 14.89131359692200, 15.45086834532493
