L(s) = 1 | + 2.19i·3-s + (−1.64 + 1.51i)5-s + i·7-s − 1.83·9-s + 1.37·11-s − 2.74i·13-s + (−3.32 − 3.61i)15-s − 6.94i·17-s + 1.29·19-s − 2.19·21-s − 8.31i·23-s + (0.412 − 4.98i)25-s + 2.56i·27-s − 8.40·29-s − 9.49·31-s + ⋯ |
L(s) = 1 | + 1.26i·3-s + (−0.735 + 0.677i)5-s + 0.377i·7-s − 0.610·9-s + 0.414·11-s − 0.760i·13-s + (−0.859 − 0.933i)15-s − 1.68i·17-s + 0.296·19-s − 0.479·21-s − 1.73i·23-s + (0.0825 − 0.996i)25-s + 0.494i·27-s − 1.56·29-s − 1.70·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7004279110\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7004279110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.64 - 1.51i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 2.19iT - 3T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2.74iT - 13T^{2} \) |
| 17 | \( 1 + 6.94iT - 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 23 | \( 1 + 8.31iT - 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 + 9.49T + 31T^{2} \) |
| 37 | \( 1 - 1.73iT - 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 + 7.83iT - 43T^{2} \) |
| 47 | \( 1 + 3.48iT - 47T^{2} \) |
| 53 | \( 1 + 6.13iT - 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 + 6.59T + 61T^{2} \) |
| 67 | \( 1 - 1.66iT - 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 2.13iT - 73T^{2} \) |
| 79 | \( 1 - 5.45T + 79T^{2} \) |
| 83 | \( 1 + 2.54iT - 83T^{2} \) |
| 89 | \( 1 + 1.43T + 89T^{2} \) |
| 97 | \( 1 - 9.69iT - 97T^{2} \) |
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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
−9.092632540550878851391409507008, −8.294234661185697684274542247784, −7.32415861538195351751065260462, −6.74276164978804470416466496039, −5.47503926270846591995834143130, −4.94506574643281477968128569425, −3.89821634569635822428741865234, −3.37097514144560132824250793252, −2.37480948731103598343784004860, −0.24824423486203386063120361449,
1.35161925946306966649220961830, 1.79746319196463507261645738383, 3.58908366439039129434071201613, 4.05411861795406416294081119160, 5.31198664060375715622025942249, 6.12647943674534126184646479119, 7.01913102104832862008689715830, 7.60294154555472733375769073620, 8.093423129038237847921783665306, 9.064413552047238055644992928040