| L(s) = 1 | + 3·3-s − 2·4-s + 5-s + 6·9-s − 6·12-s − 4·13-s + 3·15-s + 4·16-s + 2·17-s − 6·19-s − 2·20-s − 5·23-s − 4·25-s + 9·27-s − 10·29-s − 31-s − 12·36-s − 5·37-s − 12·39-s − 2·41-s + 8·43-s + 6·45-s − 8·47-s + 12·48-s + 6·51-s + 8·52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 0.447·5-s + 2·9-s − 1.73·12-s − 1.10·13-s + 0.774·15-s + 16-s + 0.485·17-s − 1.37·19-s − 0.447·20-s − 1.04·23-s − 4/5·25-s + 1.73·27-s − 1.85·29-s − 0.179·31-s − 2·36-s − 0.821·37-s − 1.92·39-s − 0.312·41-s + 1.21·43-s + 0.894·45-s − 1.16·47-s + 1.73·48-s + 0.840·51-s + 1.10·52-s − 0.824·53-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 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84049071649755321092697469836, −7.46955969774603861418264920075, −6.35024746538509655267894578430, −5.46383049703786056109210115328, −4.60641598929482242854648363332, −3.90337489403823415556541007760, −3.34496383396903427218137907992, −2.24347893763732065115208883264, −1.75713549066593075763689503387, 0,
1.75713549066593075763689503387, 2.24347893763732065115208883264, 3.34496383396903427218137907992, 3.90337489403823415556541007760, 4.60641598929482242854648363332, 5.46383049703786056109210115328, 6.35024746538509655267894578430, 7.46955969774603861418264920075, 7.84049071649755321092697469836
