L(s) = 1 | + 1.12·2-s + 3-s − 0.744·4-s + (1.27 + 1.83i)5-s + 1.12·6-s + (−2.54 + 0.709i)7-s − 3.07·8-s + 9-s + (1.42 + 2.05i)10-s + (2.57 + 2.09i)11-s − 0.744·12-s − 0.476i·13-s + (−2.85 + 0.794i)14-s + (1.27 + 1.83i)15-s − 1.95·16-s + 4.13i·17-s + ⋯ |
L(s) = 1 | + 0.792·2-s + 0.577·3-s − 0.372·4-s + (0.569 + 0.822i)5-s + 0.457·6-s + (−0.963 + 0.268i)7-s − 1.08·8-s + 0.333·9-s + (0.451 + 0.651i)10-s + (0.775 + 0.631i)11-s − 0.214·12-s − 0.132i·13-s + (−0.763 + 0.212i)14-s + (0.328 + 0.474i)15-s − 0.489·16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.099117867\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.099117867\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.27 - 1.83i)T \) |
| 7 | \( 1 + (2.54 - 0.709i)T \) |
| 11 | \( 1 + (-2.57 - 2.09i)T \) |
good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 13 | \( 1 + 0.476iT - 13T^{2} \) |
| 17 | \( 1 - 4.13iT - 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 - 2.30iT - 23T^{2} \) |
| 29 | \( 1 - 8.78iT - 29T^{2} \) |
| 31 | \( 1 + 7.12iT - 31T^{2} \) |
| 37 | \( 1 - 4.59iT - 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 + 9.14T + 47T^{2} \) |
| 53 | \( 1 - 4.46iT - 53T^{2} \) |
| 59 | \( 1 - 10.6iT - 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.29iT - 73T^{2} \) |
| 79 | \( 1 + 0.726iT - 79T^{2} \) |
| 83 | \( 1 + 8.78iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.36T + 97T^{2} \) |
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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
−9.852061239904876550861665154852, −9.315974023875276759484969684737, −8.589520261580112160182667692765, −7.36468717516655352254915433521, −6.37856262755502479042022017904, −6.01401086736066820743404194911, −4.73176421004467733376161459386, −3.70104303524099228751389591545, −3.12183579097549031089891949210, −1.93942589202562547752875518603,
0.64939761604563550980526202287, 2.41854817263728425429557909606, 3.51077807937916319641281396972, 4.25839557128757090121166951673, 5.14612189523065496423366169502, 6.16089581602501187275555685818, 6.72953907842416264192558746617, 8.185618189496018621711290163167, 8.863277113592132179197717837686, 9.497416481226136974523686554219