L(s) = 1 | + 3·5-s + 5·7-s − 5·11-s − 13-s + 8·19-s + 4·25-s + 29-s + 7·31-s + 15·35-s − 2·37-s + 10·41-s − 6·43-s − 12·47-s + 18·49-s + 6·53-s − 15·55-s + 6·59-s − 6·61-s − 3·65-s + 5·67-s − 7·71-s − 14·73-s − 25·77-s + 79-s + 83-s − 15·89-s − 5·91-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 1.88·7-s − 1.50·11-s − 0.277·13-s + 1.83·19-s + 4/5·25-s + 0.185·29-s + 1.25·31-s + 2.53·35-s − 0.328·37-s + 1.56·41-s − 0.914·43-s − 1.75·47-s + 18/7·49-s + 0.824·53-s − 2.02·55-s + 0.781·59-s − 0.768·61-s − 0.372·65-s + 0.610·67-s − 0.830·71-s − 1.63·73-s − 2.84·77-s + 0.112·79-s + 0.109·83-s − 1.58·89-s − 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.642239319\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.642239319\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 139 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.75769096028693, −16.01613197172772, −15.51537106534633, −14.73305420582626, −14.32866957151109, −13.71644812715756, −13.43437949593046, −12.69772342025317, −11.82563646460085, −11.43290994071780, −10.75534722700564, −9.974112218095449, −9.903524384971541, −8.816893275300600, −8.290343630067133, −7.623325227627101, −7.211949678494702, −6.024413599866504, −5.533810283563518, −4.947784158120465, −4.600205658841172, −3.160788723443505, −2.439703836265109, −1.767840836487812, −0.9575115747956255,
0.9575115747956255, 1.767840836487812, 2.439703836265109, 3.160788723443505, 4.600205658841172, 4.947784158120465, 5.533810283563518, 6.024413599866504, 7.211949678494702, 7.623325227627101, 8.290343630067133, 8.816893275300600, 9.903524384971541, 9.974112218095449, 10.75534722700564, 11.43290994071780, 11.82563646460085, 12.69772342025317, 13.43437949593046, 13.71644812715756, 14.32866957151109, 14.73305420582626, 15.51537106534633, 16.01613197172772, 16.75769096028693