| L(s) = 1 | + 0.596·2-s − 1.64·4-s + (−1.43 + 1.71i)5-s − 2.17·8-s + (−0.858 + 1.02i)10-s − 0.761i·11-s + 4.98·13-s + 1.99·16-s − 6.19i·17-s − 2.11i·19-s + (2.36 − 2.81i)20-s − 0.454i·22-s − 1.12·23-s + (−0.862 − 4.92i)25-s + 2.97·26-s + ⋯ |
| L(s) = 1 | + 0.421·2-s − 0.821·4-s + (−0.643 + 0.765i)5-s − 0.768·8-s + (−0.271 + 0.323i)10-s − 0.229i·11-s + 1.38·13-s + 0.497·16-s − 1.50i·17-s − 0.484i·19-s + (0.528 − 0.629i)20-s − 0.0968i·22-s − 0.234·23-s + (−0.172 − 0.985i)25-s + 0.583·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250483714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250483714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.43 - 1.71i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.596T + 2T^{2} \) |
| 11 | \( 1 + 0.761iT - 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 + 6.19iT - 17T^{2} \) |
| 19 | \( 1 + 2.11iT - 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 5.18iT - 29T^{2} \) |
| 31 | \( 1 - 7.08iT - 31T^{2} \) |
| 37 | \( 1 - 8.44iT - 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 9.44iT - 43T^{2} \) |
| 47 | \( 1 - 0.368iT - 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 2.57iT - 61T^{2} \) |
| 67 | \( 1 - 7.13iT - 67T^{2} \) |
| 71 | \( 1 - 12.4iT - 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 3.76iT - 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 + 0.0419T + 97T^{2} \) |
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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
−9.096839779024116466724851514638, −8.495038947096677084693928159414, −7.71188879803663472102649412512, −6.77077443338864311125087442113, −6.11431797701059885165338912890, −5.06507229986137728691536540105, −4.39306133481380070222477112342, −3.36161497751329989737006694550, −2.94135794678961868276723489559, −1.01069513394705770959166325556,
0.49868634296055271827868272685, 1.82264805085595880423782596642, 3.58132272658161699511234540683, 3.94832879893742843321210610681, 4.68074795297179476254273157254, 5.84585881126478805451834103592, 6.08728489201565382008383129441, 7.65842386638537590623441889309, 8.139747019955664689058357755473, 8.876653705847673583106750111282
