| L(s) = 1 | + 1.41i·3-s − 5-s − 3.46·7-s + 0.999·9-s + (−1.73 + 2.82i)11-s − 2.44i·13-s − 1.41i·15-s − 7.34i·17-s + 3.46·19-s − 4.89i·21-s − 1.41i·23-s + 25-s + 5.65i·27-s + 4.89i·29-s + (−4.00 − 2.44i)33-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s − 0.447·5-s − 1.30·7-s + 0.333·9-s + (−0.522 + 0.852i)11-s − 0.679i·13-s − 0.365i·15-s − 1.78i·17-s + 0.794·19-s − 1.06i·21-s − 0.294i·23-s + 0.200·25-s + 1.08i·27-s + 0.909i·29-s + (−0.696 − 0.426i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260749565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260749565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (1.73 - 2.82i)T \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 7.34iT - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.992629208647820455160543660707, −7.81196853408339624621479909943, −7.17951062001512309003105953683, −6.66975363260945261244683227051, −5.36001927509572987511825025854, −4.98120333149918520377607383622, −3.97691220765139072369452001491, −3.25246068053860745480637933703, −2.53002291305404684763322371042, −0.72907177810745207428870289810,
0.58620024585478860802486636401, 1.76433113869434993150920635163, 2.90017139116790947350834982148, 3.68733665016043380924599906664, 4.44505659455528952989783985087, 5.88418731196876359320579169527, 6.15104433939302532207300998807, 6.99995339384840205702447327152, 7.67317177834380688454631138732, 8.285413298386001419090746467827
