| L(s) = 1 | + (−2 + 2.23i)3-s − 2.23i·5-s − 2·7-s + (−1.00 − 8.94i)9-s − 13.4i·11-s + 8·13-s + (5.00 + 4.47i)15-s + 13.4i·17-s + 34·19-s + (4 − 4.47i)21-s − 40.2i·23-s − 5.00·25-s + (22.0 + 15.6i)27-s − 40.2i·29-s − 14·31-s + ⋯ |
| L(s) = 1 | + (−0.666 + 0.745i)3-s − 0.447i·5-s − 0.285·7-s + (−0.111 − 0.993i)9-s − 1.21i·11-s + 0.615·13-s + (0.333 + 0.298i)15-s + 0.789i·17-s + 1.78·19-s + (0.190 − 0.212i)21-s − 1.74i·23-s − 0.200·25-s + (0.814 + 0.579i)27-s − 1.38i·29-s − 0.451·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04361 - 0.398624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04361 - 0.398624i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2 - 2.23i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 13.4iT - 121T^{2} \) |
| 13 | \( 1 - 8T + 169T^{2} \) |
| 17 | \( 1 - 13.4iT - 289T^{2} \) |
| 19 | \( 1 - 34T + 361T^{2} \) |
| 23 | \( 1 + 40.2iT - 529T^{2} \) |
| 29 | \( 1 + 40.2iT - 841T^{2} \) |
| 31 | \( 1 + 14T + 961T^{2} \) |
| 37 | \( 1 - 56T + 1.36e3T^{2} \) |
| 41 | \( 1 + 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 40.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 40.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 13.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 46T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32T + 4.48e3T^{2} \) |
| 71 | \( 1 + 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106T + 5.32e3T^{2} \) |
| 79 | \( 1 - 22T + 6.24e3T^{2} \) |
| 83 | \( 1 - 120. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 122T + 9.40e3T^{2} \) |
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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
−11.61463529123126255680878263803, −10.91809737861071506096622982979, −9.910082970761950223351205460749, −9.005573745742292200058942456802, −8.025773303184323858916193764338, −6.33933711002365588561088727038, −5.68422698109558736905296363283, −4.39793582494968958516912021908, −3.25400737175870062306895408962, −0.72093817613293607114905671851,
1.45434820455017486105777129164, 3.14551828905361041951591584773, 4.92482270008296073106892430765, 5.94632777445677102714009961557, 7.19655001376276928028719173631, 7.55759314885657683149523693394, 9.267594117484508364717482258331, 10.13884717696738528414571348664, 11.35444133131044377004341116583, 11.84635163800132264612389136374
