| L(s) = 1 | − 2.23·2-s + 3.00·4-s − 1.23·5-s + 3.23·7-s − 2.23·8-s − 3·9-s + 2.76·10-s − 0.763·11-s + 13-s − 7.23·14-s − 0.999·16-s + 2·17-s + 6.70·18-s + 7.23·19-s − 3.70·20-s + 1.70·22-s − 23-s − 3.47·25-s − 2.23·26-s + 9.70·28-s + 8.47·29-s + 10.4·31-s + 6.70·32-s − 4.47·34-s − 4.00·35-s − 9.00·36-s − 1.23·37-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.552·5-s + 1.22·7-s − 0.790·8-s − 9-s + 0.874·10-s − 0.230·11-s + 0.277·13-s − 1.93·14-s − 0.249·16-s + 0.485·17-s + 1.58·18-s + 1.66·19-s − 0.829·20-s + 0.364·22-s − 0.208·23-s − 0.694·25-s − 0.438·26-s + 1.83·28-s + 1.57·29-s + 1.88·31-s + 1.18·32-s − 0.766·34-s − 0.676·35-s − 1.50·36-s − 0.203·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6053827217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6053827217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 0.472T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−11.56606705836910880022757088016, −10.74505833984061487029852009881, −9.785787410312295043018323834255, −8.740170974942945239697434964312, −7.984673280684728591471738415464, −7.56931325630531505993436971978, −6.01908499747306097706202056858, −4.66218077370200323094439355473, −2.79161701957007952815446190995, −1.06886688320617586041121682212,
1.06886688320617586041121682212, 2.79161701957007952815446190995, 4.66218077370200323094439355473, 6.01908499747306097706202056858, 7.56931325630531505993436971978, 7.984673280684728591471738415464, 8.740170974942945239697434964312, 9.785787410312295043018323834255, 10.74505833984061487029852009881, 11.56606705836910880022757088016
