| L(s) = 1 | + (−1.48 + 1.67i)5-s − 0.806i·7-s + 11-s − 2.15i·13-s + 2.54i·17-s + 0.387·19-s + (−0.612 − 4.96i)25-s − 0.649·29-s − 4.96·31-s + (1.35 + 1.19i)35-s − 8.31i·37-s + 2.57·41-s + 0.806i·43-s − 1.61i·47-s + 6.35·49-s + ⋯ |
| L(s) = 1 | + (−0.662 + 0.749i)5-s − 0.304i·7-s + 0.301·11-s − 0.598i·13-s + 0.617i·17-s + 0.0889·19-s + (−0.122 − 0.992i)25-s − 0.120·29-s − 0.891·31-s + (0.228 + 0.201i)35-s − 1.36i·37-s + 0.402·41-s + 0.122i·43-s − 0.235i·47-s + 0.907·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328780112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328780112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.48 - 1.67i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 0.806iT - 7T^{2} \) |
| 13 | \( 1 + 2.15iT - 13T^{2} \) |
| 17 | \( 1 - 2.54iT - 17T^{2} \) |
| 19 | \( 1 - 0.387T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 0.649T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 8.31iT - 37T^{2} \) |
| 41 | \( 1 - 2.57T + 41T^{2} \) |
| 43 | \( 1 - 0.806iT - 43T^{2} \) |
| 47 | \( 1 + 1.61iT - 47T^{2} \) |
| 53 | \( 1 + 0.649iT - 53T^{2} \) |
| 59 | \( 1 + 0.649T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.22iT - 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 - 0.231iT - 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + 4.08iT - 83T^{2} \) |
| 89 | \( 1 - 4.88T + 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−8.300619869525827167422648735934, −7.49360812902968019601627651602, −7.12025757518638308247896285041, −6.16950303703722986683957659284, −5.52131563365679899323424929709, −4.36755713742361140147646805301, −3.75182817350942971734839457904, −2.99279226181585655969390146757, −1.91487941403682733760385099362, −0.48678756386793239930030384512,
0.908211194320517475548899421632, 2.00936543981977051858977189747, 3.17528270631638288823377506234, 4.02665370140555710969453826940, 4.75780066247846898622091394497, 5.44391679698015372271050570936, 6.36337637072957191409205898995, 7.18587430024700143159514257431, 7.81830974237169641753753870784, 8.629218900745324072506938948162
