L(s) = 1 | − 3-s − 2·4-s − 5-s − 7-s + 9-s + 2·11-s + 2·12-s + 13-s + 15-s + 4·16-s + 2·17-s + 4·19-s + 2·20-s + 21-s + 25-s − 27-s + 2·28-s − 5·29-s + 2·31-s − 2·33-s + 35-s − 2·36-s − 39-s − 6·41-s − 9·43-s − 4·44-s − 45-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 0.277·13-s + 0.258·15-s + 16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.928·29-s + 0.359·31-s − 0.348·33-s + 0.169·35-s − 1/3·36-s − 0.160·39-s − 0.937·41-s − 1.37·43-s − 0.603·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143745 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143745 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.54749519657554, −13.08661691589965, −12.79950573637408, −12.04267727962434, −11.70160355491908, −11.53946401001712, −10.56851106596752, −10.30238818441700, −9.752279806436214, −9.306658596012715, −8.861227372277235, −8.309070692400743, −7.727152526088367, −7.380343029857200, −6.483579349125151, −6.396366759787737, −5.426395647304859, −5.228003343438328, −4.667421258516521, −3.940545864968926, −3.530091024079487, −3.207767149365801, −2.093991928811507, −1.289837084124148, −0.7449675614520871, 0,
0.7449675614520871, 1.289837084124148, 2.093991928811507, 3.207767149365801, 3.530091024079487, 3.940545864968926, 4.667421258516521, 5.228003343438328, 5.426395647304859, 6.396366759787737, 6.483579349125151, 7.380343029857200, 7.727152526088367, 8.309070692400743, 8.861227372277235, 9.306658596012715, 9.752279806436214, 10.30238818441700, 10.56851106596752, 11.53946401001712, 11.70160355491908, 12.04267727962434, 12.79950573637408, 13.08661691589965, 13.54749519657554