| L(s) = 1 | + (1 − i)2-s − 2i·4-s + (−3 + 3i)7-s + (−2 − 2i)8-s − 12·11-s + (−12 − 12i)13-s + 6i·14-s − 4·16-s + (−12 + 12i)17-s − 20i·19-s + (−12 + 12i)22-s + (−3 − 3i)23-s − 24·26-s + (6 + 6i)28-s − 30i·29-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.5i)2-s − 0.5i·4-s + (−0.428 + 0.428i)7-s + (−0.250 − 0.250i)8-s − 1.09·11-s + (−0.923 − 0.923i)13-s + 0.428i·14-s − 0.250·16-s + (−0.705 + 0.705i)17-s − 1.05i·19-s + (−0.545 + 0.545i)22-s + (−0.130 − 0.130i)23-s − 0.923·26-s + (0.214 + 0.214i)28-s − 1.03i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5043332159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5043332159\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (3 - 3i)T - 49iT^{2} \) |
| 11 | \( 1 + 12T + 121T^{2} \) |
| 13 | \( 1 + (12 + 12i)T + 169iT^{2} \) |
| 17 | \( 1 + (12 - 12i)T - 289iT^{2} \) |
| 19 | \( 1 + 20iT - 361T^{2} \) |
| 23 | \( 1 + (3 + 3i)T + 529iT^{2} \) |
| 29 | \( 1 + 30iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 + (48 - 48i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 48T + 1.68e3T^{2} \) |
| 43 | \( 1 + (27 + 27i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (27 - 27i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-12 - 12i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 60iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32T + 3.72e3T^{2} \) |
| 67 | \( 1 + (3 - 3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 48T + 5.04e3T^{2} \) |
| 73 | \( 1 + (12 + 12i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 40iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (93 + 93i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 30iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-12 + 12i)T - 9.40e3iT^{2} \) |
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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
−10.44294953053361651290031001357, −9.760960734492452382314169658146, −8.662657052204385408560293276179, −7.63620326212253654900255948705, −6.46147206179342856635772207486, −5.43532237680396697126432624571, −4.59503965952966961387836556509, −3.12150638235640145834482265512, −2.27355510390479788293446068657, −0.16075624130832180634033572950,
2.26501707871681033292559392503, 3.59399985154746245982779657036, 4.72762904327044007051422065485, 5.61385385054849307645092436142, 6.88165710361960955613939425367, 7.39623409825557126083905220068, 8.531020441407849239558909366870, 9.564319432652994993661748681734, 10.44045677273259063978480810578, 11.45978368462385252032605115576
