| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 4i·7-s − i·8-s − 9-s − 11-s + i·12-s + 4·14-s + 16-s − 8i·17-s − i·18-s − 19-s − 4·21-s − i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 0.301·11-s + 0.288i·12-s + 1.06·14-s + 0.250·16-s − 1.94i·17-s − 0.235i·18-s − 0.229·19-s − 0.872·21-s − 0.213i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7072944890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7072944890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 11iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.131249668215418664036003275647, −7.58917468173779138562205069234, −6.99714789244424905392739362598, −6.49482679930193600594389305397, −5.34297716034151522974190983796, −4.72034007583467629561633064723, −3.75229426427201101216240727280, −2.78031562473522690595690063204, −1.27978972478432578826574830260, −0.23279685588340032062251358667,
1.71613523417364555227614327199, 2.53489072780661097603399962009, 3.43047708817711198218963468974, 4.30351690166654931702167538274, 5.20778686681708888727461539453, 5.83854768007935758634163650597, 6.60348240883292456273867449923, 8.050809400799723817092333708446, 8.565921616009679837369866078455, 9.019547245185164822498182028485
