| L(s) = 1 | − 2.82·3-s + 2·5-s − 4.24·7-s + 5.00·9-s − 1.41·11-s − 13-s − 5.65·15-s − 4·17-s − 1.41·19-s + 12·21-s + 5.65·23-s − 25-s − 5.65·27-s − 2·29-s − 9.89·31-s + 4.00·33-s − 8.48·35-s + 2·37-s + 2.82·39-s − 6·41-s + 11.3·43-s + 10.0·45-s − 7.07·47-s + 10.9·49-s + 11.3·51-s + 12·53-s − 2.82·55-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.894·5-s − 1.60·7-s + 1.66·9-s − 0.426·11-s − 0.277·13-s − 1.46·15-s − 0.970·17-s − 0.324·19-s + 2.61·21-s + 1.17·23-s − 0.200·25-s − 1.08·27-s − 0.371·29-s − 1.77·31-s + 0.696·33-s − 1.43·35-s + 0.328·37-s + 0.452·39-s − 0.937·41-s + 1.72·43-s + 1.49·45-s − 1.03·47-s + 1.57·49-s + 1.58·51-s + 1.64·53-s − 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5017062864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5017062864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.099995699930779623772441215276, −7.50826545566524521058591693287, −6.82265295508250819198289295974, −6.29298758819490773401862171239, −5.71074544360407297487807844017, −5.10429215164250801281554736911, −4.12909147352484610836893804000, −3.00436263079979411666230178844, −1.90473663130062823693360957355, −0.43898740622011626404420918963,
0.43898740622011626404420918963, 1.90473663130062823693360957355, 3.00436263079979411666230178844, 4.12909147352484610836893804000, 5.10429215164250801281554736911, 5.71074544360407297487807844017, 6.29298758819490773401862171239, 6.82265295508250819198289295974, 7.50826545566524521058591693287, 9.099995699930779623772441215276
