| L(s) = 1 | + 1.05e4·5-s − 4.93e4·7-s − 3.09e5·11-s − 1.72e6·13-s + 2.27e6·17-s − 4.55e6·19-s − 7.28e6·23-s + 6.20e7·25-s + 6.90e7·29-s + 1.41e8·31-s − 5.19e8·35-s + 7.11e8·37-s + 1.22e9·41-s + 3.36e7·43-s + 1.23e8·47-s + 4.53e8·49-s − 1.10e9·53-s − 3.25e9·55-s − 9.06e9·59-s − 3.85e9·61-s − 1.81e10·65-s + 1.53e10·67-s + 2.06e10·71-s − 2.06e9·73-s + 1.52e10·77-s − 1.36e10·79-s + 6.55e10·83-s + ⋯ |
| L(s) = 1 | + 1.50·5-s − 1.10·7-s − 0.579·11-s − 1.28·13-s + 0.389·17-s − 0.421·19-s − 0.235·23-s + 1.27·25-s + 0.625·29-s + 0.889·31-s − 1.67·35-s + 1.68·37-s + 1.65·41-s + 0.0348·43-s + 0.0783·47-s + 0.229·49-s − 0.363·53-s − 0.872·55-s − 1.65·59-s − 0.584·61-s − 1.94·65-s + 1.38·67-s + 1.35·71-s − 0.116·73-s + 0.642·77-s − 0.500·79-s + 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.191455900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191455900\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2106 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 49304 T + p^{11} T^{2} \) |
| 11 | \( 1 + 309420 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1723594 T + p^{11} T^{2} \) |
| 17 | \( 1 - 2279502 T + p^{11} T^{2} \) |
| 19 | \( 1 + 4550444 T + p^{11} T^{2} \) |
| 23 | \( 1 + 7282872 T + p^{11} T^{2} \) |
| 29 | \( 1 - 69040026 T + p^{11} T^{2} \) |
| 31 | \( 1 - 141740704 T + p^{11} T^{2} \) |
| 37 | \( 1 - 711366974 T + p^{11} T^{2} \) |
| 41 | \( 1 - 1225262214 T + p^{11} T^{2} \) |
| 43 | \( 1 - 781540 p T + p^{11} T^{2} \) |
| 47 | \( 1 - 123214608 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1106121582 T + p^{11} T^{2} \) |
| 59 | \( 1 + 9062779932 T + p^{11} T^{2} \) |
| 61 | \( 1 + 3854150458 T + p^{11} T^{2} \) |
| 67 | \( 1 - 15313764676 T + p^{11} T^{2} \) |
| 71 | \( 1 - 20619626328 T + p^{11} T^{2} \) |
| 73 | \( 1 + 2063718694 T + p^{11} T^{2} \) |
| 79 | \( 1 + 13689871472 T + p^{11} T^{2} \) |
| 83 | \( 1 - 65570428908 T + p^{11} T^{2} \) |
| 89 | \( 1 - 29715508854 T + p^{11} T^{2} \) |
| 97 | \( 1 + 23439626206 T + p^{11} T^{2} \) |
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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
−10.69920141914488943945203612020, −9.773618141634740882663229575592, −9.434128301892921206914789096610, −7.84727725903804805051336904847, −6.54640020498057524168613408692, −5.83173849890209507944403678114, −4.67087506248656714218727822133, −2.91497648241681979933729892707, −2.21003512010945897253659736452, −0.67726835440130799454953771944,
0.67726835440130799454953771944, 2.21003512010945897253659736452, 2.91497648241681979933729892707, 4.67087506248656714218727822133, 5.83173849890209507944403678114, 6.54640020498057524168613408692, 7.84727725903804805051336904847, 9.434128301892921206914789096610, 9.773618141634740882663229575592, 10.69920141914488943945203612020
