L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 4·11-s + 12-s − 13-s + 16-s + 6·17-s − 18-s + 4·19-s − 4·22-s − 8·23-s − 24-s + 26-s + 27-s + 6·29-s − 8·31-s − 32-s + 4·33-s − 6·34-s + 36-s + 10·37-s − 4·38-s − 39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.752923080\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752923080\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.322358641232610436784648285374, −8.325976297604312961734027796655, −7.83466570589962625459131201889, −7.01490063638720124480711152469, −6.20008378576346310195929413718, −5.25724539674339433994151433277, −3.96841057609218045594012616188, −3.26819059787837421750330634066, −2.05799963788222426145971930454, −1.02048172072810586725444806438,
1.02048172072810586725444806438, 2.05799963788222426145971930454, 3.26819059787837421750330634066, 3.96841057609218045594012616188, 5.25724539674339433994151433277, 6.20008378576346310195929413718, 7.01490063638720124480711152469, 7.83466570589962625459131201889, 8.325976297604312961734027796655, 9.322358641232610436784648285374