| L(s) = 1 | − 2-s − 3-s + 4-s + 2.49·5-s + 6-s − 8-s + 9-s − 2.49·10-s − 3.46·11-s − 12-s − 4.98·13-s − 2.49·15-s + 16-s + 2.24·17-s − 18-s + 19-s + 2.49·20-s + 3.46·22-s + 6.00·23-s + 24-s + 1.21·25-s + 4.98·26-s − 27-s − 5.51·29-s + 2.49·30-s − 7.28·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.11·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.788·10-s − 1.04·11-s − 0.288·12-s − 1.38·13-s − 0.643·15-s + 0.250·16-s + 0.544·17-s − 0.235·18-s + 0.229·19-s + 0.557·20-s + 0.738·22-s + 1.25·23-s + 0.204·24-s + 0.242·25-s + 0.977·26-s − 0.192·27-s − 1.02·29-s + 0.455·30-s − 1.30·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050059500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050059500\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.49T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 + 5.87T + 53T^{2} \) |
| 59 | \( 1 + 8.67T + 59T^{2} \) |
| 61 | \( 1 - 4.37T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−8.020758640646063974376171629995, −7.35520867859197461360649381908, −6.93713274929115785005196719710, −5.85059592794873442722624307545, −5.41460955225856559116393703934, −4.87453076045653853893701419983, −3.48866677007819485940150159308, −2.46452221540714075445432646705, −1.87325107224201813687647184864, −0.60887240030285224411112307281,
0.60887240030285224411112307281, 1.87325107224201813687647184864, 2.46452221540714075445432646705, 3.48866677007819485940150159308, 4.87453076045653853893701419983, 5.41460955225856559116393703934, 5.85059592794873442722624307545, 6.93713274929115785005196719710, 7.35520867859197461360649381908, 8.020758640646063974376171629995
