L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s − 7-s + 8-s + 9-s + 4·10-s − 3·11-s + 12-s − 13-s − 14-s + 4·15-s + 16-s − 3·17-s + 18-s + 8·19-s + 4·20-s − 21-s − 3·22-s − 2·23-s + 24-s + 11·25-s − 26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.83·19-s + 0.894·20-s − 0.218·21-s − 0.639·22-s − 0.417·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.837335956\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.837335956\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.77895253297084242187803238126, −6.80077319423564236576591922362, −6.55045818335262538818149330619, −5.50640811468805468124684080899, −5.24397343535953275419212171421, −4.41750350347952329249166475356, −3.21924584421368170058873532014, −2.70765641825735211321579171363, −2.10461974585332099041513973501, −1.09461925745493623715747023130,
1.09461925745493623715747023130, 2.10461974585332099041513973501, 2.70765641825735211321579171363, 3.21924584421368170058873532014, 4.41750350347952329249166475356, 5.24397343535953275419212171421, 5.50640811468805468124684080899, 6.55045818335262538818149330619, 6.80077319423564236576591922362, 7.77895253297084242187803238126