| L(s) = 1 | + 2.52·2-s + 0.384i·3-s + 4.39·4-s − 5-s + 0.973i·6-s + 1.35i·7-s + 6.07·8-s + 2.85·9-s − 2.52·10-s + (1.95 + 2.68i)11-s + 1.69i·12-s − 2.33·13-s + 3.42i·14-s − 0.384i·15-s + 6.55·16-s − 5.43i·17-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.222i·3-s + 2.19·4-s − 0.447·5-s + 0.397i·6-s + 0.511i·7-s + 2.14·8-s + 0.950·9-s − 0.799·10-s + (0.587 + 0.808i)11-s + 0.488i·12-s − 0.648·13-s + 0.914i·14-s − 0.0993i·15-s + 1.63·16-s − 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.614958522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.614958522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + (-1.95 - 2.68i)T \) |
| 19 | \( 1 + (0.531 - 4.32i)T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 3 | \( 1 - 0.384iT - 3T^{2} \) |
| 7 | \( 1 - 1.35iT - 7T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 + 5.43iT - 17T^{2} \) |
| 23 | \( 1 - 0.981T + 23T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 + 8.09iT - 31T^{2} \) |
| 37 | \( 1 - 1.40iT - 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 2.01iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 3.95iT - 53T^{2} \) |
| 59 | \( 1 - 1.26iT - 59T^{2} \) |
| 61 | \( 1 + 9.55iT - 61T^{2} \) |
| 67 | \( 1 + 13.2iT - 67T^{2} \) |
| 71 | \( 1 - 14.3iT - 71T^{2} \) |
| 73 | \( 1 + 2.05iT - 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 12.3iT - 89T^{2} \) |
| 97 | \( 1 + 2.23iT - 97T^{2} \) |
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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
−10.03595047430290891359799751811, −9.451613502912905019939299060831, −7.955172233348238599652874797602, −7.11829660163671023772313740245, −6.51597064588394280609667728308, −5.36873895304933565092912236649, −4.63899153976569623534116017981, −4.03076009520499503775488257661, −2.96471231718745197169009167387, −1.86253027382964460545243948599,
1.43036918028311174743598377707, 2.86753591657325142330182456959, 3.83958558011231258054320742631, 4.43899753937317125327641142274, 5.34409401443120969522951447993, 6.53513866796123864080522573759, 6.87614676877847614649255201145, 7.82373572377857195710839946571, 8.922492983761849530885425543339, 10.33007619796402524279202592720
