| L(s) = 1 | + 0.317i·3-s + 2.89·9-s + 3.78i·11-s + 1.89·17-s + 5.97i·19-s + 1.87i·27-s − 1.20·33-s − 6.79·41-s − 8.48i·43-s − 7·49-s + 0.603i·51-s − 1.89·57-s + 14.1i·59-s + 16.3i·67-s + 15.6·73-s + ⋯ |
| L(s) = 1 | + 0.183i·3-s + 0.966·9-s + 1.14i·11-s + 0.460·17-s + 1.37i·19-s + 0.360i·27-s − 0.209·33-s − 1.06·41-s − 1.29i·43-s − 49-s + 0.0845i·51-s − 0.251·57-s + 1.84i·59-s + 1.99i·67-s + 1.83·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760272664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760272664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.317iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 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 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 16.3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 17.0iT - 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.889786572883767456920230322597, −8.015990568919252171893311842259, −7.34004554813528478615838108404, −6.75486194952162976879698813938, −5.76718477696392616676563607441, −4.95762288531116673030986609288, −4.16670901801288976727145591440, −3.47641566163354016037920353399, −2.15792020858860269943023734412, −1.32375590742799428679056681339,
0.57128140795724239532338152535, 1.67845939906725181624099824744, 2.89140266603604311305672398330, 3.66421006401019417141165307053, 4.69243630231824038262369112038, 5.34781597212091731735130827024, 6.46409983543521407964348469810, 6.80216703826085228549045146696, 7.890042634629577817163274401356, 8.287198514017754229458927179117
