L(s) = 1 | − 2-s + 2.21·3-s + 4-s − 3.55·5-s − 2.21·6-s − 4.06·7-s − 8-s + 1.91·9-s + 3.55·10-s − 1.35·11-s + 2.21·12-s − 3.28·13-s + 4.06·14-s − 7.88·15-s + 16-s − 3.40·17-s − 1.91·18-s + 3.01·19-s − 3.55·20-s − 9.00·21-s + 1.35·22-s + 23-s − 2.21·24-s + 7.65·25-s + 3.28·26-s − 2.41·27-s − 4.06·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.27·3-s + 0.5·4-s − 1.59·5-s − 0.904·6-s − 1.53·7-s − 0.353·8-s + 0.636·9-s + 1.12·10-s − 0.408·11-s + 0.639·12-s − 0.909·13-s + 1.08·14-s − 2.03·15-s + 0.250·16-s − 0.826·17-s − 0.450·18-s + 0.691·19-s − 0.795·20-s − 1.96·21-s + 0.288·22-s + 0.208·23-s − 0.452·24-s + 1.53·25-s + 0.643·26-s − 0.464·27-s − 0.768·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4045662971\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4045662971\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 + 6.00T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 6.99T + 67T^{2} \) |
| 71 | \( 1 + 3.90T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 1.87T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.108887504325294934644745250865, −7.44141460841764232663697059412, −7.12774024847232818445168732022, −6.30199040194182998872441325461, −5.08277896110022637329152539857, −4.08923755741862106075360033261, −3.23316770526391664746086941256, −3.07982405814059398849953166746, −2.02325013267179049263782711654, −0.32189093371020734712945477328,
0.32189093371020734712945477328, 2.02325013267179049263782711654, 3.07982405814059398849953166746, 3.23316770526391664746086941256, 4.08923755741862106075360033261, 5.08277896110022637329152539857, 6.30199040194182998872441325461, 7.12774024847232818445168732022, 7.44141460841764232663697059412, 8.108887504325294934644745250865