L(s) = 1 | + (0.5 + 0.866i)7-s − 1.73i·13-s + (−1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 + 0.866i)79-s + (1.49 − 0.866i)91-s + (1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)7-s − 1.73i·13-s + (−1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 + 0.866i)79-s + (1.49 − 0.866i)91-s + (1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7533523450\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7533523450\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 1.73iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−12.53215418346690045431137012082, −11.24298370669853202270486110399, −10.53267159925723775319435664971, −9.356016630938252112091714826295, −8.344641000529020091597113578714, −7.58550144724325126417600350942, −5.99967589066269309991929784482, −5.29873520295787314240505170371, −3.72761380043161358497732490050, −2.18384318450784974480618441631,
1.98126574448115951940535471276, 3.95066593784166164853909707177, 4.78770231110273723801441485478, 6.43820644479442424950699200760, 7.19442353313728619277730897997, 8.434738428211970939090069030582, 9.311494397191903195635700960849, 10.53020636809064090326126324008, 11.20793083446542881049647399051, 12.17999464888387279957474132642