L(s) = 1 | + (−0.0770 − 1.41i)2-s + 3-s + (−1.98 + 0.217i)4-s + (−2.13 − 0.658i)5-s + (−0.0770 − 1.41i)6-s + (−3.54 − 3.54i)7-s + (0.460 + 2.79i)8-s + 9-s + (−0.765 + 3.06i)10-s + (−0.707 + 0.707i)11-s + (−1.98 + 0.217i)12-s − 1.18i·13-s + (−4.73 + 5.28i)14-s + (−2.13 − 0.658i)15-s + (3.90 − 0.865i)16-s + (−2.63 − 2.63i)17-s + ⋯ |
L(s) = 1 | + (−0.0544 − 0.998i)2-s + 0.577·3-s + (−0.994 + 0.108i)4-s + (−0.955 − 0.294i)5-s + (−0.0314 − 0.576i)6-s + (−1.34 − 1.34i)7-s + (0.162 + 0.986i)8-s + 0.333·9-s + (−0.242 + 0.970i)10-s + (−0.213 + 0.213i)11-s + (−0.573 + 0.0628i)12-s − 0.329i·13-s + (−1.26 + 1.41i)14-s + (−0.551 − 0.170i)15-s + (0.976 − 0.216i)16-s + (−0.639 − 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0297794 - 0.744901i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0297794 - 0.744901i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0770 + 1.41i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (2.13 + 0.658i)T \) |
good | 7 | \( 1 + (3.54 + 3.54i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.18iT - 13T^{2} \) |
| 17 | \( 1 + (2.63 + 2.63i)T + 17iT^{2} \) |
| 19 | \( 1 + (-5.21 + 5.21i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.86 - 1.86i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.17 - 2.17i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 0.910iT - 37T^{2} \) |
| 41 | \( 1 - 8.26iT - 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (-5.06 + 5.06i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + (10.2 + 10.2i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.49 + 4.49i)T - 61iT^{2} \) |
| 67 | \( 1 - 1.27iT - 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 + (2.47 + 2.47i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.89T + 79T^{2} \) |
| 83 | \( 1 - 9.99T + 83T^{2} \) |
| 89 | \( 1 + 5.16T + 89T^{2} \) |
| 97 | \( 1 + (6.87 + 6.87i)T + 97iT^{2} \) |
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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
−11.65361869209411162039738352716, −10.68041745409888035621524120716, −9.769545719260021479981182734322, −9.033783429874416643914276659267, −7.76644350385258187475288765437, −6.98279921954791062169166727154, −4.85385470168996600027912368633, −3.78687757957935059674054295600, −2.97267014213894544179917428637, −0.58175488167602208106421502868,
2.99745562218334358860100560501, 4.09066397900404834375519269116, 5.72164992443394212526232026478, 6.59898261467646176637880711384, 7.71766289689317600644811454044, 8.603173017501750967370975132017, 9.347576755233487110439695701687, 10.35834024130781490955262797915, 12.02915244848368297496735077185, 12.60393953570839058699305300003