| L(s) = 1 | + 3-s − 3.78i·5-s − 3.68·7-s + 9-s + (0.386 + 3.29i)11-s − 5.10·13-s − 3.78i·15-s − 1.27i·17-s + 6.09i·19-s − 3.68·21-s − 2.32i·23-s − 9.34·25-s + 27-s + 3.97·29-s + 8.34i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.69i·5-s − 1.39·7-s + 0.333·9-s + (0.116 + 0.993i)11-s − 1.41·13-s − 0.978i·15-s − 0.308i·17-s + 1.39i·19-s − 0.804·21-s − 0.484i·23-s − 1.86·25-s + 0.192·27-s + 0.737·29-s + 1.49i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4073088358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4073088358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + (-0.386 - 3.29i)T \) |
| good | 5 | \( 1 + 3.78iT - 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 1.27iT - 17T^{2} \) |
| 19 | \( 1 - 6.09iT - 19T^{2} \) |
| 23 | \( 1 + 2.32iT - 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 8.34iT - 31T^{2} \) |
| 37 | \( 1 - 6.23iT - 37T^{2} \) |
| 41 | \( 1 + 4.90iT - 41T^{2} \) |
| 43 | \( 1 + 8.64iT - 43T^{2} \) |
| 47 | \( 1 + 3.01iT - 47T^{2} \) |
| 53 | \( 1 - 9.33iT - 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 0.803T + 67T^{2} \) |
| 71 | \( 1 - 8.56iT - 71T^{2} \) |
| 73 | \( 1 - 5.39iT - 73T^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 2.02T + 97T^{2} \) |
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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
−9.225643679657575413438246732738, −8.789391128309897387665012853260, −7.83807292834400407094592106220, −7.13772069005093313151701268665, −6.22423768465277340264960949239, −5.11854947108242854721517214449, −4.57467170950389669216434890372, −3.61352776296001076412199120813, −2.51865724202708376317864680436, −1.37874131703543262733005452557,
0.12746344917904255709670693020, 2.36144455959744617947632031189, 2.99236149255235475149890327949, 3.46489699951318754436010249885, 4.68585383610772693385841121773, 6.22650806158979008055143499706, 6.34961119679991062988433931308, 7.38187356300904225749166701012, 7.77932738131477396062702576823, 9.176775422743166597300908564449
