| L(s) = 1 | + 2.77·3-s + 2.31·7-s + 4.67·9-s − 2.16·11-s − 6.25·13-s − 7.10·17-s + 19-s + 6.40·21-s − 8.99·23-s + 4.63·27-s + 3.16·29-s + 9.95·31-s − 5.98·33-s − 9.43·37-s − 17.3·39-s − 10.8·41-s + 1.05·43-s − 6.98·47-s − 1.66·49-s − 19.6·51-s + 2.69·53-s + 2.77·57-s − 11.2·59-s − 4.36·61-s + 10.7·63-s + 6.25·67-s − 24.9·69-s + ⋯ |
| L(s) = 1 | + 1.59·3-s + 0.873·7-s + 1.55·9-s − 0.651·11-s − 1.73·13-s − 1.72·17-s + 0.229·19-s + 1.39·21-s − 1.87·23-s + 0.892·27-s + 0.587·29-s + 1.78·31-s − 1.04·33-s − 1.55·37-s − 2.77·39-s − 1.68·41-s + 0.160·43-s − 1.01·47-s − 0.237·49-s − 2.75·51-s + 0.370·53-s + 0.366·57-s − 1.45·59-s − 0.558·61-s + 1.36·63-s + 0.764·67-s − 3.00·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 23 | \( 1 + 8.99T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 - 9.95T + 31T^{2} \) |
| 37 | \( 1 + 9.43T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 4.36T + 61T^{2} \) |
| 67 | \( 1 - 6.25T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.64T + 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−7.82910337621629357278985768467, −7.03446551605195146612841303297, −6.38863326200938573989075656354, −4.96809267323638294905636588761, −4.77590696491682074753368837203, −3.86882505245860189016958756184, −2.92454298028065355397106180435, −2.23200998671857837340803908899, −1.82529571928063356520382359159, 0,
1.82529571928063356520382359159, 2.23200998671857837340803908899, 2.92454298028065355397106180435, 3.86882505245860189016958756184, 4.77590696491682074753368837203, 4.96809267323638294905636588761, 6.38863326200938573989075656354, 7.03446551605195146612841303297, 7.82910337621629357278985768467
