L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·13-s + 14-s + 16-s − 2·17-s − 20-s + 4·23-s + 25-s − 2·26-s − 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s + 35-s + 2·37-s + 40-s − 2·41-s + 4·43-s − 4·46-s + 4·47-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.328·37-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.589·46-s + 0.583·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.13170520981213, −12.66051408899272, −12.21946781659044, −11.55085452410858, −11.34830284135962, −10.87938579416999, −10.26353817914121, −10.02450082711602, −9.306278799809454, −8.811219878226818, −8.745934613571542, −7.891540473179405, −7.606567234343673, −7.063886375516832, −6.592138217759356, −6.022922396604931, −5.711184299815840, −4.792286317268179, −4.472230024561113, −3.734033073464149, −3.236226113637453, −2.715822348200201, −2.070017927396039, −1.326524295141274, −0.7400825798078273, 0,
0.7400825798078273, 1.326524295141274, 2.070017927396039, 2.715822348200201, 3.236226113637453, 3.734033073464149, 4.472230024561113, 4.792286317268179, 5.711184299815840, 6.022922396604931, 6.592138217759356, 7.063886375516832, 7.606567234343673, 7.891540473179405, 8.745934613571542, 8.811219878226818, 9.306278799809454, 10.02450082711602, 10.26353817914121, 10.87938579416999, 11.34830284135962, 11.55085452410858, 12.21946781659044, 12.66051408899272, 13.13170520981213