L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s − 4·11-s + 12-s + 2·13-s − 4·14-s + 16-s − 17-s − 18-s − 4·19-s + 4·21-s + 4·22-s + 4·23-s − 24-s − 2·26-s + 27-s + 4·28-s + 2·29-s + 4·31-s − 32-s − 4·33-s + 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.872·21-s + 0.852·22-s + 0.834·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.696·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.899299959\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.899299959\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.627802270178912501969894591143, −8.242236607138952274114988738210, −7.70103286147411178889763109774, −6.87077232056441766899307255120, −5.82653552853551697654708741657, −4.90751585366850821424912904843, −4.15332031001663255062607373177, −2.79629461207611571430929335087, −2.11061561108870647278195413477, −0.981756754426503792719893088438,
0.981756754426503792719893088438, 2.11061561108870647278195413477, 2.79629461207611571430929335087, 4.15332031001663255062607373177, 4.90751585366850821424912904843, 5.82653552853551697654708741657, 6.87077232056441766899307255120, 7.70103286147411178889763109774, 8.242236607138952274114988738210, 8.627802270178912501969894591143