| L(s) = 1 | + (−3.67 − 3.67i)7-s + (8.57 − 8.57i)13-s + 11i·19-s − 59·31-s + (−48.9 − 48.9i)37-s + (15.9 − 15.9i)43-s − 22i·49-s − 121·61-s + (−94.3 − 94.3i)67-s + (97.9 − 97.9i)73-s + 142i·79-s − 63.0·91-s + (−69.8 − 69.8i)97-s + (48.9 − 48.9i)103-s − 71i·109-s + ⋯ |
| L(s) = 1 | + (−0.524 − 0.524i)7-s + (0.659 − 0.659i)13-s + 0.578i·19-s − 1.90·31-s + (−1.32 − 1.32i)37-s + (0.370 − 0.370i)43-s − 0.448i·49-s − 1.98·61-s + (−1.40 − 1.40i)67-s + (1.34 − 1.34i)73-s + 1.79i·79-s − 0.692·91-s + (−0.719 − 0.719i)97-s + (0.475 − 0.475i)103-s − 0.651i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7550713197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7550713197\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (3.67 + 3.67i)T + 49iT^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + (-8.57 + 8.57i)T - 169iT^{2} \) |
| 17 | \( 1 + 289iT^{2} \) |
| 19 | \( 1 - 11iT - 361T^{2} \) |
| 23 | \( 1 - 529iT^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 59T + 961T^{2} \) |
| 37 | \( 1 + (48.9 + 48.9i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + (-15.9 + 15.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 2.20e3iT^{2} \) |
| 53 | \( 1 - 2.80e3iT^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 121T + 3.72e3T^{2} \) |
| 67 | \( 1 + (94.3 + 94.3i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 5.04e3T^{2} \) |
| 73 | \( 1 + (-97.9 + 97.9i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 142iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3iT^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + (69.8 + 69.8i)T + 9.40e3iT^{2} \) |
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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
−9.552475003207565453761169544754, −8.825050394219006884561525861679, −7.80880138890182193567022905255, −7.08615564304192185531023825050, −6.07585617912797048323443751675, −5.28910394697055141403280144449, −3.94955016084030417726320252155, −3.27278600232603447281446337769, −1.74645330303596515819919544647, −0.24018541084053786089295201078,
1.56098341207091176266503495583, 2.86129759598374418450847519364, 3.87150818030317918221971778075, 5.00958930613954862110099710877, 5.99340645474011640701789010567, 6.75235049888331846035026291465, 7.67594721839216985737082842724, 8.866048050536118957887177299151, 9.168440808806188326098699891564, 10.24998596309792271472088008697
