| L(s) = 1 | − 1.52·2-s + 3·3-s − 5.67·4-s + 9.65·5-s − 4.57·6-s + 22.3·7-s + 20.8·8-s + 9·9-s − 14.7·10-s − 50.3·11-s − 17.0·12-s − 34.0·14-s + 28.9·15-s + 13.6·16-s + 86.1·17-s − 13.7·18-s + 116.·19-s − 54.8·20-s + 67.0·21-s + 76.6·22-s + 72·23-s + 62.5·24-s − 31.7·25-s + 27·27-s − 126.·28-s + 14.1·29-s − 44.1·30-s + ⋯ |
| L(s) = 1 | − 0.538·2-s + 0.577·3-s − 0.709·4-s + 0.863·5-s − 0.310·6-s + 1.20·7-s + 0.921·8-s + 0.333·9-s − 0.465·10-s − 1.37·11-s − 0.409·12-s − 0.650·14-s + 0.498·15-s + 0.213·16-s + 1.22·17-s − 0.179·18-s + 1.41·19-s − 0.613·20-s + 0.697·21-s + 0.742·22-s + 0.652·23-s + 0.531·24-s − 0.253·25-s + 0.192·27-s − 0.857·28-s + 0.0905·29-s − 0.268·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.080738443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080738443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 5 | \( 1 - 9.65T + 125T^{2} \) |
| 7 | \( 1 - 22.3T + 343T^{2} \) |
| 11 | \( 1 + 50.3T + 1.33e3T^{2} \) |
| 17 | \( 1 - 86.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 329.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 768.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 771.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 527.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 74.2T + 9.12e5T^{2} \) |
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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
−10.20327347432945256538524816535, −9.639950297496626953603760935423, −8.708095747635906666889706621148, −7.902395662415847262566855081575, −7.36929001614901920528939392001, −5.29142048183080041743039120322, −5.22585386927545034833024039883, −3.57427478172441228488495301152, −2.14080464143328436142059398934, −1.02026820438467910550872509312,
1.02026820438467910550872509312, 2.14080464143328436142059398934, 3.57427478172441228488495301152, 5.22585386927545034833024039883, 5.29142048183080041743039120322, 7.36929001614901920528939392001, 7.902395662415847262566855081575, 8.708095747635906666889706621148, 9.639950297496626953603760935423, 10.20327347432945256538524816535
