| L(s) = 1 | − 3-s − 2·4-s − 3·5-s − 2·9-s + 2·12-s − 4·13-s + 3·15-s + 4·16-s − 6·17-s + 2·19-s + 6·20-s + 3·23-s + 4·25-s + 5·27-s + 6·29-s − 5·31-s + 4·36-s + 11·37-s + 4·39-s + 6·41-s − 8·43-s + 6·45-s − 4·48-s + 6·51-s + 8·52-s − 6·53-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s + 0.577·12-s − 1.10·13-s + 0.774·15-s + 16-s − 1.45·17-s + 0.458·19-s + 1.34·20-s + 0.625·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s − 0.898·31-s + 2/3·36-s + 1.80·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.894·45-s − 0.577·48-s + 0.840·51-s + 1.10·52-s − 0.824·53-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.965599239814975840296884001805, −7.05102888697694657596792316990, −6.35946393928988064816172947330, −5.33457363303989968032950767365, −4.77627871815223419331599351974, −4.26760634035281585123892442853, −3.37588925165810734532381222346, −2.51077378003816655346374440626, −0.77148952963174533854789924573, 0,
0.77148952963174533854789924573, 2.51077378003816655346374440626, 3.37588925165810734532381222346, 4.26760634035281585123892442853, 4.77627871815223419331599351974, 5.33457363303989968032950767365, 6.35946393928988064816172947330, 7.05102888697694657596792316990, 7.965599239814975840296884001805
