| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.239190006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239190006\) |
| \(L(2)\) |
|
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
−10.07258019724007816794038757490, −9.364310726698541293408478130025, −8.509945586934526144594692220790, −7.69717659040541656337595681803, −6.80347442840356855242904826097, −5.77084916069425994504066038358, −4.93191754337461095511987842696, −3.93703906137588490072620364432, −2.62004166930718335417139326822, −1.53269928442286524882833473546,
0.39390715347581614843637532209, 1.89233153118569605740593524478, 3.16140343884655656541631163970, 4.31299743052100206375649048772, 5.17240798948554430240164045333, 6.13131315599137126714387429400, 7.38371155081238627897199074767, 7.69360031945845897560042183246, 8.895828530255645839856346496382, 9.591402527881157524836142003327
