L(s) = 1 | + 2.97·3-s + 4.29i·5-s + 2.64i·7-s + 5.82·9-s − 0.737i·13-s + 12.7i·15-s + 5.43·19-s + 7.86i·21-s − 7.48i·23-s − 13.4·25-s + 8.40·27-s − 11.3·35-s − 2.19i·39-s + 25.0i·45-s − 7.00·49-s + ⋯ |
L(s) = 1 | + 1.71·3-s + 1.92i·5-s + 0.999i·7-s + 1.94·9-s − 0.204i·13-s + 3.29i·15-s + 1.24·19-s + 1.71i·21-s − 1.56i·23-s − 2.69·25-s + 1.61·27-s − 1.92·35-s − 0.350i·39-s + 3.73i·45-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.263596351\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.263596351\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 - 4.29iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 0.737iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 23 | \( 1 + 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.45T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.568342859241745646133180322949, −8.559531836401813859964668325020, −8.006934958170694978279504645006, −7.17270402553896955660716086765, −6.60107771026634160166014110154, −5.55199644202201308678503050667, −4.11720908732470432025275307726, −3.13378362240551053303312175677, −2.77878772948080805674055621091, −2.00501283909746227249308836956,
1.04628088297666194502980030677, 1.79697438066615160203114329376, 3.27192836174525397274491024192, 3.99112206991323480395948132801, 4.72163125184598627652589259433, 5.63651310704775191864978457804, 7.19319153841218235610179471367, 7.72122650694593350952267488283, 8.331113352560482452035066856865, 9.117608680321717648873554275933