| L(s) = 1 | − 2·5-s + 11-s − 2·13-s − 17-s + 3·19-s + 23-s − 25-s − 29-s − 2·31-s + 5·37-s + 10·41-s − 43-s − 7·47-s + 12·53-s − 2·55-s − 3·59-s + 14·61-s + 4·65-s − 12·67-s − 5·71-s − 8·73-s − 6·83-s + 2·85-s + 6·89-s − 6·95-s + 7·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s − 0.554·13-s − 0.242·17-s + 0.688·19-s + 0.208·23-s − 1/5·25-s − 0.185·29-s − 0.359·31-s + 0.821·37-s + 1.56·41-s − 0.152·43-s − 1.02·47-s + 1.64·53-s − 0.269·55-s − 0.390·59-s + 1.79·61-s + 0.496·65-s − 1.46·67-s − 0.593·71-s − 0.936·73-s − 0.658·83-s + 0.216·85-s + 0.635·89-s − 0.615·95-s + 0.710·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.518626057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.518626057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.77267173890564, −11.99517001004206, −11.78917842622117, −11.37510782393364, −10.97342654187519, −10.37058237776564, −9.845524281449499, −9.527608029685781, −8.850505420822209, −8.626253681610097, −7.857641493385636, −7.574086414494424, −7.211416141614993, −6.674275638051251, −6.030349692258423, −5.606206869765412, −5.021812207659495, −4.439766711957029, −4.060273582887017, −3.569099782309386, −2.920990011603220, −2.460843256380992, −1.726808845948516, −1.008697825178974, −0.3722241621173979,
0.3722241621173979, 1.008697825178974, 1.726808845948516, 2.460843256380992, 2.920990011603220, 3.569099782309386, 4.060273582887017, 4.439766711957029, 5.021812207659495, 5.606206869765412, 6.030349692258423, 6.674275638051251, 7.211416141614993, 7.574086414494424, 7.857641493385636, 8.626253681610097, 8.850505420822209, 9.527608029685781, 9.845524281449499, 10.37058237776564, 10.97342654187519, 11.37510782393364, 11.78917842622117, 11.99517001004206, 12.77267173890564
