| L(s) = 1 | + 0.317·3-s − 2.89·9-s + 3.78i·11-s + 1.89i·17-s − 5.97i·19-s − 1.87·27-s + 1.20i·33-s − 6.79·41-s − 8.48·43-s + 7·49-s + 0.603i·51-s − 1.89i·57-s − 14.1i·59-s − 16.3·67-s − 15.6i·73-s + ⋯ |
| L(s) = 1 | + 0.183·3-s − 0.966·9-s + 1.14i·11-s + 0.460i·17-s − 1.37i·19-s − 0.360·27-s + 0.209i·33-s − 1.06·41-s − 1.29·43-s + 49-s + 0.0845i·51-s − 0.251i·57-s − 1.84i·59-s − 1.99·67-s − 1.83i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2317442926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2317442926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.317T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.89iT - 17T^{2} \) |
| 19 | \( 1 + 5.97iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 16.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 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
−8.460602547901092457342711849296, −7.58877683597671246775380799231, −6.89008148694733605725965057351, −6.14700381639684445313610324206, −5.16967276121894812213395564487, −4.57945231574847202829901531909, −3.49316291295560357471349108757, −2.63692331324075320031265688784, −1.71312540201654906595398209230, −0.06695356096495571207658588099,
1.37070245846762250700265940440, 2.67591513594408314304671028176, 3.34091203388788965352702552212, 4.21136351492394598985681736918, 5.46171770321994109174046145133, 5.77690416367500846876460013364, 6.70225012914020184956702911575, 7.61091142923431111636320411084, 8.463416211059999353741812491823, 8.681703893546581773666194276250
