| L(s) = 1 | + 2-s + 4-s − 2.66·5-s − 3.18·7-s + 8-s − 2.66·10-s − 5.51·11-s + 2.64·13-s − 3.18·14-s + 16-s + 3.74·17-s − 6.37·19-s − 2.66·20-s − 5.51·22-s + 2.12·25-s + 2.64·26-s − 3.18·28-s − 5.50·29-s − 10.6·31-s + 32-s + 3.74·34-s + 8.50·35-s + 9.36·37-s − 6.37·38-s − 2.66·40-s − 0.424·41-s − 5.38·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.19·5-s − 1.20·7-s + 0.353·8-s − 0.844·10-s − 1.66·11-s + 0.734·13-s − 0.851·14-s + 0.250·16-s + 0.908·17-s − 1.46·19-s − 0.596·20-s − 1.17·22-s + 0.425·25-s + 0.519·26-s − 0.602·28-s − 1.02·29-s − 1.91·31-s + 0.176·32-s + 0.642·34-s + 1.43·35-s + 1.53·37-s − 1.03·38-s − 0.422·40-s − 0.0663·41-s − 0.820·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9168205680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9168205680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 9.36T + 37T^{2} \) |
| 41 | \( 1 + 0.424T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 9.38T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 - 1.91T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 - 5.15T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.68236515757224182067389961443, −7.06299946659149639563774921642, −6.14279711901069429900911000794, −5.73969811072831733324403121470, −4.87160896372136815335195676895, −4.02938657988074226774013946491, −3.51147846783036449766067470368, −2.93060248462468296332025140028, −1.97239454487837072215266614547, −0.38297803759806851401536172757,
0.38297803759806851401536172757, 1.97239454487837072215266614547, 2.93060248462468296332025140028, 3.51147846783036449766067470368, 4.02938657988074226774013946491, 4.87160896372136815335195676895, 5.73969811072831733324403121470, 6.14279711901069429900911000794, 7.06299946659149639563774921642, 7.68236515757224182067389961443
