| L(s) = 1 | − 9.11·5-s − 11·11-s − 43.3·23-s + 58.1·25-s − 61.5·31-s − 47.8·37-s + 50·47-s + 49·49-s + 70·53-s + 100.·55-s − 10.4·59-s − 94.5·67-s + 109.·71-s + 80.7·89-s + 98.9·97-s + 190·103-s + 167.·113-s + 395.·115-s + ⋯ |
| L(s) = 1 | − 1.82·5-s − 11-s − 1.88·23-s + 2.32·25-s − 1.98·31-s − 1.29·37-s + 1.06·47-s + 0.999·49-s + 1.32·53-s + 1.82·55-s − 0.176·59-s − 1.41·67-s + 1.54·71-s + 0.907·89-s + 1.02·97-s + 1.84·103-s + 1.48·113-s + 3.43·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6131995393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6131995393\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 9.11T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 43.3T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 61.5T + 961T^{2} \) |
| 37 | \( 1 + 47.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 50T + 2.20e3T^{2} \) |
| 53 | \( 1 - 70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 94.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 109.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 98.9T + 9.40e3T^{2} \) |
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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
−8.979141358046744401718523136416, −8.352866878490961205059824350128, −7.53655142305454357701561520426, −7.24633686536194978258263392308, −5.91071065390175569022347336533, −4.99372139788301903002411934845, −4.00853351250419451658636899497, −3.46506909828205863418179698408, −2.17918240927245145478945171632, −0.41823043148460933934833604088,
0.41823043148460933934833604088, 2.17918240927245145478945171632, 3.46506909828205863418179698408, 4.00853351250419451658636899497, 4.99372139788301903002411934845, 5.91071065390175569022347336533, 7.24633686536194978258263392308, 7.53655142305454357701561520426, 8.352866878490961205059824350128, 8.979141358046744401718523136416
