L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 4·13-s + 15-s − 17-s − 8·19-s − 23-s + 25-s + 27-s − 5·29-s − 4·31-s + 33-s − 2·37-s + 4·39-s − 10·41-s − 8·43-s + 45-s − 9·47-s − 51-s − 4·53-s + 55-s − 8·57-s + 6·61-s + 4·65-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.242·17-s − 1.83·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s − 1.56·41-s − 1.21·43-s + 0.149·45-s − 1.31·47-s − 0.140·51-s − 0.549·53-s + 0.134·55-s − 1.05·57-s + 0.768·61-s + 0.496·65-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.378764649\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.378764649\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.37446643615750, −13.01006943476000, −12.73981675229629, −12.06981886246382, −11.35162996847122, −11.08800088929738, −10.55177258291451, −9.976181306189585, −9.624560946652404, −8.905415736698770, −8.623382277069230, −8.233921010687856, −7.673844397821391, −6.835564933370432, −6.526458184049598, −6.199776047256917, −5.324966170674427, −4.981765868881962, −4.134708439111327, −3.725908045336386, −3.297716350270534, −2.438252805884617, −1.764937400618197, −1.602740317559966, −0.4146186451512535,
0.4146186451512535, 1.602740317559966, 1.764937400618197, 2.438252805884617, 3.297716350270534, 3.725908045336386, 4.134708439111327, 4.981765868881962, 5.324966170674427, 6.199776047256917, 6.526458184049598, 6.835564933370432, 7.673844397821391, 8.233921010687856, 8.623382277069230, 8.905415736698770, 9.624560946652404, 9.976181306189585, 10.55177258291451, 11.08800088929738, 11.35162996847122, 12.06981886246382, 12.73981675229629, 13.01006943476000, 13.37446643615750