L(s) = 1 | − 0.897·3-s − 0.897·5-s + 7-s − 2.19·9-s − 5.85·11-s − 4.66·13-s + 0.805·15-s − 7.35·17-s + 4·19-s − 0.897·21-s − 23-s − 4.19·25-s + 4.66·27-s + 10.5·29-s + 7.49·31-s + 5.25·33-s − 0.897·35-s + 8.25·37-s + 4.18·39-s − 2·41-s + 6.66·43-s + 1.96·45-s − 0.692·47-s + 49-s + 6.60·51-s − 12.8·53-s + 5.25·55-s + ⋯ |
L(s) = 1 | − 0.518·3-s − 0.401·5-s + 0.377·7-s − 0.731·9-s − 1.76·11-s − 1.29·13-s + 0.207·15-s − 1.78·17-s + 0.917·19-s − 0.195·21-s − 0.208·23-s − 0.838·25-s + 0.897·27-s + 1.95·29-s + 1.34·31-s + 0.914·33-s − 0.151·35-s + 1.35·37-s + 0.669·39-s − 0.312·41-s + 1.01·43-s + 0.293·45-s − 0.101·47-s + 0.142·49-s + 0.924·51-s − 1.76·53-s + 0.708·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6847743985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6847743985\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 0.897T + 3T^{2} \) |
| 5 | \( 1 + 0.897T + 5T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 8.25T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 + 0.692T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.28T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 + 6.15T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−8.682543019462509502094187689019, −8.084161170546760420149036510476, −7.48250634959240407031212716016, −6.55793475739089268339041760673, −5.70036976073875285343281811006, −4.85692179326725026275709951284, −4.48589690447663746588702749400, −2.83246588508938092518459194932, −2.42679368150241520026847150077, −0.50041465073087556366078482450,
0.50041465073087556366078482450, 2.42679368150241520026847150077, 2.83246588508938092518459194932, 4.48589690447663746588702749400, 4.85692179326725026275709951284, 5.70036976073875285343281811006, 6.55793475739089268339041760673, 7.48250634959240407031212716016, 8.084161170546760420149036510476, 8.682543019462509502094187689019