L(s) = 1 | + 2.15·2-s + 3-s + 2.63·4-s + (1.03 + 1.98i)5-s + 2.15·6-s + (2.36 − 1.18i)7-s + 1.35·8-s + 9-s + (2.23 + 4.26i)10-s + (2.31 − 2.37i)11-s + 2.63·12-s − 2.85i·13-s + (5.09 − 2.54i)14-s + (1.03 + 1.98i)15-s − 2.33·16-s + 4.53i·17-s + ⋯ |
L(s) = 1 | + 1.52·2-s + 0.577·3-s + 1.31·4-s + (0.463 + 0.885i)5-s + 0.878·6-s + (0.894 − 0.447i)7-s + 0.480·8-s + 0.333·9-s + (0.705 + 1.34i)10-s + (0.697 − 0.716i)11-s + 0.759·12-s − 0.791i·13-s + (1.36 − 0.680i)14-s + (0.267 + 0.511i)15-s − 0.584·16-s + 1.10i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.095529459\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.095529459\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (-1.03 - 1.98i)T \) |
| 7 | \( 1 + (-2.36 + 1.18i)T \) |
| 11 | \( 1 + (-2.31 + 2.37i)T \) |
good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 13 | \( 1 + 2.85iT - 13T^{2} \) |
| 17 | \( 1 - 4.53iT - 17T^{2} \) |
| 19 | \( 1 + 8.19T + 19T^{2} \) |
| 23 | \( 1 + 3.73iT - 23T^{2} \) |
| 29 | \( 1 + 2.78iT - 29T^{2} \) |
| 31 | \( 1 - 8.30iT - 31T^{2} \) |
| 37 | \( 1 - 2.54iT - 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 2.80T + 47T^{2} \) |
| 53 | \( 1 + 3.66iT - 53T^{2} \) |
| 59 | \( 1 + 2.02iT - 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 13.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 6.09iT - 73T^{2} \) |
| 79 | \( 1 - 9.72iT - 79T^{2} \) |
| 83 | \( 1 + 1.96iT - 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−10.20582119230357758037736608977, −8.732413468844813996782563365920, −8.207743681531692401093354703058, −6.93211066467644017471499523750, −6.36759843422142243042431032917, −5.54403402791604083972742877036, −4.39878755075222486542596546337, −3.74975133923296595125017192223, −2.81397800750975401105670595542, −1.79971368746681203616183050025,
1.78426432619487153499368212784, 2.42836705360643567104621511011, 4.01616098707102011610357114441, 4.48107165814204574433192706144, 5.24232048296894251595583758005, 6.17007561478053497261961492216, 7.04895715153115797206824858148, 8.132912364508812125686934851598, 9.131619392371505189760080744024, 9.414492378369774519097845164845