L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 2·11-s − 12-s − 2·13-s − 14-s − 15-s + 16-s + 2·17-s − 18-s + 2·19-s + 20-s − 21-s − 2·22-s + 8·23-s + 24-s + 25-s + 2·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.218·21-s − 0.426·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.79246470450190, −12.28485900165435, −12.05957211690661, −11.49318578763881, −11.07752264805726, −10.66549205850622, −10.17874666254263, −9.738027501308808, −9.304700128759986, −8.930803006851112, −8.309245355487008, −7.875467439694820, −7.316718717352310, −6.772983775492743, −6.605060082116117, −5.875415448917614, −5.423169288617308, −4.848514788011975, −4.584833637029935, −3.694124939575675, −3.068266731493912, −2.657812679446573, −1.816599007037165, −1.296072204888332, −0.8902010391524320, 0,
0.8902010391524320, 1.296072204888332, 1.816599007037165, 2.657812679446573, 3.068266731493912, 3.694124939575675, 4.584833637029935, 4.848514788011975, 5.423169288617308, 5.875415448917614, 6.605060082116117, 6.772983775492743, 7.316718717352310, 7.875467439694820, 8.309245355487008, 8.930803006851112, 9.304700128759986, 9.738027501308808, 10.17874666254263, 10.66549205850622, 11.07752264805726, 11.49318578763881, 12.05957211690661, 12.28485900165435, 12.79246470450190