| L(s) = 1 | − 0.820·2-s + 2.32·3-s − 1.32·4-s − 1.90·6-s + 0.561·7-s + 2.72·8-s + 2.41·9-s − 0.111·11-s − 3.08·12-s − 6.89·13-s − 0.460·14-s + 0.415·16-s + 3.81·17-s − 1.98·18-s + 1.30·21-s + 0.0911·22-s + 4.04·23-s + 6.35·24-s + 5.65·26-s − 1.35·27-s − 0.745·28-s + 9.29·29-s − 4.18·31-s − 5.79·32-s − 0.258·33-s − 3.12·34-s − 3.20·36-s + ⋯ |
| L(s) = 1 | − 0.580·2-s + 1.34·3-s − 0.663·4-s − 0.779·6-s + 0.212·7-s + 0.964·8-s + 0.805·9-s − 0.0334·11-s − 0.891·12-s − 1.91·13-s − 0.123·14-s + 0.103·16-s + 0.924·17-s − 0.467·18-s + 0.285·21-s + 0.0194·22-s + 0.842·23-s + 1.29·24-s + 1.10·26-s − 0.261·27-s − 0.140·28-s + 1.72·29-s − 0.752·31-s − 1.02·32-s − 0.0450·33-s − 0.536·34-s − 0.534·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.858494892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858494892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.820T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 11 | \( 1 + 0.111T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.63T + 43T^{2} \) |
| 47 | \( 1 + 2.12T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.64T + 71T^{2} \) |
| 73 | \( 1 - 8.45T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 - 8.16T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−7.899948557137795700534979329526, −7.36652139881291832003816791643, −6.77406625437381102559314008420, −5.34090141603202049402666747276, −4.99185325500268495975495569105, −4.13718582523150664112369315228, −3.32235907851241905727257435317, −2.61373096439204824221474921813, −1.80510993150680264316845239973, −0.67687070821380005541659113017,
0.67687070821380005541659113017, 1.80510993150680264316845239973, 2.61373096439204824221474921813, 3.32235907851241905727257435317, 4.13718582523150664112369315228, 4.99185325500268495975495569105, 5.34090141603202049402666747276, 6.77406625437381102559314008420, 7.36652139881291832003816791643, 7.899948557137795700534979329526
