| L(s) = 1 | − 0.414·3-s − 2.82·9-s − 3.82·11-s + 3.58·13-s + 6.41·17-s − 3.65·19-s − 0.585·23-s + 2.41·27-s + 6.65·29-s − 4.58·31-s + 1.58·33-s + 3.41·37-s − 1.48·39-s − 0.585·41-s − 11.6·43-s − 8.89·47-s − 2.65·51-s + 3.75·53-s + 1.51·57-s − 3.41·59-s − 5.17·61-s + 11.0·67-s + 0.242·69-s + 6.48·71-s + 5.17·73-s + 13.1·79-s + 7.48·81-s + ⋯ |
| L(s) = 1 | − 0.239·3-s − 0.942·9-s − 1.15·11-s + 0.994·13-s + 1.55·17-s − 0.838·19-s − 0.122·23-s + 0.464·27-s + 1.23·29-s − 0.823·31-s + 0.276·33-s + 0.561·37-s − 0.237·39-s − 0.0914·41-s − 1.77·43-s − 1.29·47-s − 0.372·51-s + 0.516·53-s + 0.200·57-s − 0.444·59-s − 0.662·61-s + 1.35·67-s + 0.0292·69-s + 0.769·71-s + 0.605·73-s + 1.47·79-s + 0.831·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.361161527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.361161527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 - 6.41T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 0.585T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.266715791654007284199784674159, −7.74753664414478560233940379182, −6.67334678594833714094697425699, −6.03747783793947854682941272301, −5.38306296317573869575170057496, −4.75545191256132167349950433369, −3.53915601125648699072611097541, −3.00871417529586453069695919571, −1.91982331486241435031840599837, −0.63636941224015383187025527762,
0.63636941224015383187025527762, 1.91982331486241435031840599837, 3.00871417529586453069695919571, 3.53915601125648699072611097541, 4.75545191256132167349950433369, 5.38306296317573869575170057496, 6.03747783793947854682941272301, 6.67334678594833714094697425699, 7.74753664414478560233940379182, 8.266715791654007284199784674159
