| L(s) = 1 | − 0.600·3-s + 2.23·5-s − 0.532·7-s − 2.63·9-s + 5.09·11-s − 3.68·13-s − 1.34·15-s − 0.993·17-s − 2.48·19-s + 0.319·21-s + 2.14·23-s + 0.00667·25-s + 3.38·27-s − 9.53·29-s + 0.838·31-s − 3.05·33-s − 1.19·35-s − 1.17·37-s + 2.20·39-s − 3.98·41-s − 3.86·43-s − 5.90·45-s + 4.30·47-s − 6.71·49-s + 0.596·51-s − 0.989·53-s + 11.3·55-s + ⋯ |
| L(s) = 1 | − 0.346·3-s + 1.00·5-s − 0.201·7-s − 0.879·9-s + 1.53·11-s − 1.02·13-s − 0.346·15-s − 0.241·17-s − 0.571·19-s + 0.0697·21-s + 0.447·23-s + 0.00133·25-s + 0.651·27-s − 1.77·29-s + 0.150·31-s − 0.531·33-s − 0.201·35-s − 0.193·37-s + 0.353·39-s − 0.622·41-s − 0.589·43-s − 0.880·45-s + 0.628·47-s − 0.959·49-s + 0.0835·51-s − 0.135·53-s + 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.600T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 + 0.532T + 7T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 + 0.993T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 9.53T + 29T^{2} \) |
| 31 | \( 1 - 0.838T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 - 4.30T + 47T^{2} \) |
| 53 | \( 1 + 0.989T + 53T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 0.877T + 67T^{2} \) |
| 71 | \( 1 + 2.80T + 71T^{2} \) |
| 73 | \( 1 + 1.96T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 - 8.01T + 97T^{2} \) |
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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
−8.187728855361090262894572042073, −7.08713760226579202959436764238, −6.56516994768956547587079117360, −5.85023864002231551845514628092, −5.27767150161072196777589402115, −4.33126583270701809886694624304, −3.37401747657471847259398509579, −2.35539121261291524789932581362, −1.52302152439771810471341643329, 0,
1.52302152439771810471341643329, 2.35539121261291524789932581362, 3.37401747657471847259398509579, 4.33126583270701809886694624304, 5.27767150161072196777589402115, 5.85023864002231551845514628092, 6.56516994768956547587079117360, 7.08713760226579202959436764238, 8.187728855361090262894572042073
