L(s) = 1 | − 14·3-s − 158·7-s − 47·9-s + 148·11-s + 684·13-s + 2.04e3·17-s − 2.22e3·19-s + 2.21e3·21-s + 1.24e3·23-s + 4.06e3·27-s − 270·29-s + 2.04e3·31-s − 2.07e3·33-s − 4.37e3·37-s − 9.57e3·39-s − 2.39e3·41-s − 2.29e3·43-s + 1.06e4·47-s + 8.15e3·49-s − 2.86e4·51-s + 2.96e3·53-s + 3.10e4·57-s + 3.97e4·59-s − 4.22e4·61-s + 7.42e3·63-s − 3.20e4·67-s − 1.74e4·69-s + ⋯ |
L(s) = 1 | − 0.898·3-s − 1.21·7-s − 0.193·9-s + 0.368·11-s + 1.12·13-s + 1.71·17-s − 1.41·19-s + 1.09·21-s + 0.491·23-s + 1.07·27-s − 0.0596·29-s + 0.382·31-s − 0.331·33-s − 0.525·37-s − 1.00·39-s − 0.222·41-s − 0.189·43-s + 0.705·47-s + 0.485·49-s − 1.54·51-s + 0.144·53-s + 1.26·57-s + 1.48·59-s − 1.45·61-s + 0.235·63-s − 0.873·67-s − 0.441·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 14 T + p^{5} T^{2} \) |
| 7 | \( 1 + 158 T + p^{5} T^{2} \) |
| 11 | \( 1 - 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 684 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2048 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2220 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1246 T + p^{5} T^{2} \) |
| 29 | \( 1 + 270 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2048 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4372 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2398 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2294 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10682 T + p^{5} T^{2} \) |
| 53 | \( 1 - 2964 T + p^{5} T^{2} \) |
| 59 | \( 1 - 39740 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42298 T + p^{5} T^{2} \) |
| 67 | \( 1 + 32098 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 30104 T + p^{5} T^{2} \) |
| 79 | \( 1 + 35280 T + p^{5} T^{2} \) |
| 83 | \( 1 - 27826 T + p^{5} T^{2} \) |
| 89 | \( 1 + 85210 T + p^{5} T^{2} \) |
| 97 | \( 1 + 97232 T + p^{5} T^{2} \) |
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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
−10.20771487315585414587680164957, −9.148233438416766643585784698501, −8.232861870289449805047603775330, −6.82149096000923941535880579112, −6.16924931379894897809447463669, −5.42333451135211666453339371535, −3.95948350904642629795907165015, −2.97405388990275610712631840378, −1.15330240094540768562011900902, 0,
1.15330240094540768562011900902, 2.97405388990275610712631840378, 3.95948350904642629795907165015, 5.42333451135211666453339371535, 6.16924931379894897809447463669, 6.82149096000923941535880579112, 8.232861870289449805047603775330, 9.148233438416766643585784698501, 10.20771487315585414587680164957