| L(s) = 1 | + 2.41·2-s − 2.82·3-s + 3.82·4-s − 5-s − 6.82·6-s − 2·7-s + 4.41·8-s + 5.00·9-s − 2.41·10-s + 11-s − 10.8·12-s − 1.17·13-s − 4.82·14-s + 2.82·15-s + 2.99·16-s + 6.82·17-s + 12.0·18-s − 3.82·20-s + 5.65·21-s + 2.41·22-s − 2.82·23-s − 12.4·24-s + 25-s − 2.82·26-s − 5.65·27-s − 7.65·28-s − 3.65·29-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 1.63·3-s + 1.91·4-s − 0.447·5-s − 2.78·6-s − 0.755·7-s + 1.56·8-s + 1.66·9-s − 0.763·10-s + 0.301·11-s − 3.12·12-s − 0.324·13-s − 1.29·14-s + 0.730·15-s + 0.749·16-s + 1.65·17-s + 2.84·18-s − 0.856·20-s + 1.23·21-s + 0.514·22-s − 0.589·23-s − 2.54·24-s + 0.200·25-s − 0.554·26-s − 1.08·27-s − 1.44·28-s − 0.679·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.197056967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197056967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−15.34960948708537437517013199535, −14.10678218029280958479780612667, −12.69871244487788176118942150891, −12.18106320759827562104741174082, −11.34576766095860460302436742769, −10.09200070663283635358165303574, −7.19557878469241887119343312891, −6.08854885975527628228900382813, −5.16733014703725630411671370919, −3.70347734990671907639955801345,
3.70347734990671907639955801345, 5.16733014703725630411671370919, 6.08854885975527628228900382813, 7.19557878469241887119343312891, 10.09200070663283635358165303574, 11.34576766095860460302436742769, 12.18106320759827562104741174082, 12.69871244487788176118942150891, 14.10678218029280958479780612667, 15.34960948708537437517013199535
