L(s) = 1 | + 2i·5-s − 4i·13-s + 8·17-s + 25-s + 10i·29-s + 12i·37-s + 8·41-s − 7·49-s + 14i·53-s − 12i·61-s + 8·65-s − 6·73-s + 16i·85-s + 16·89-s + 18·97-s + ⋯ |
L(s) = 1 | + 0.894i·5-s − 1.10i·13-s + 1.94·17-s + 0.200·25-s + 1.85i·29-s + 1.97i·37-s + 1.24·41-s − 49-s + 1.92i·53-s − 1.53i·61-s + 0.992·65-s − 0.702·73-s + 1.73i·85-s + 1.69·89-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.675345593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.675345593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 12iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−10.13852419384693859791938650779, −9.154023296156999509157275982202, −8.054031608146775213709927224735, −7.52058281928250969497260568069, −6.57431551751645437142274452492, −5.70426406218297600320158904448, −4.85520788150980356817474621153, −3.33587521588229330439508665186, −2.96975377663197412870246570883, −1.23654910196504229459775032339,
0.883939896211164024581406165787, 2.17637646862648988184829450490, 3.62316112592333676467084540495, 4.48912698965887130754000845781, 5.43560605668167663455910920234, 6.20441728924070196660451788414, 7.38636052496249254937044044400, 8.032695675415850048715968186992, 8.992160693593646143980419697927, 9.569195906209353316386103391816