| L(s) = 1 | + 1.55·3-s + 1.73·7-s − 0.590·9-s − 4.43·11-s + 0.878·13-s − 17-s + 1.18·19-s + 2.69·21-s + 4.73·23-s − 5.57·27-s + 1.18·29-s − 8.89·31-s − 6.87·33-s − 5.18·37-s + 1.36·39-s − 9.12·41-s + 1.12·43-s + 5.16·47-s − 3.98·49-s − 1.55·51-s + 2.56·53-s + 1.83·57-s − 12.3·59-s − 7.85·61-s − 1.02·63-s + 5.98·67-s + 7.34·69-s + ⋯ |
| L(s) = 1 | + 0.896·3-s + 0.656·7-s − 0.196·9-s − 1.33·11-s + 0.243·13-s − 0.242·17-s + 0.270·19-s + 0.588·21-s + 0.986·23-s − 1.07·27-s + 0.219·29-s − 1.59·31-s − 1.19·33-s − 0.851·37-s + 0.218·39-s − 1.42·41-s + 0.171·43-s + 0.753·47-s − 0.569·49-s − 0.217·51-s + 0.352·53-s + 0.242·57-s − 1.60·59-s − 1.00·61-s − 0.129·63-s + 0.730·67-s + 0.884·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 - 0.878T + 13T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 - 2.56T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 - 5.98T + 67T^{2} \) |
| 71 | \( 1 - 6.18T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 8.07T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−7.70375403356815760643800507915, −7.22304148114232549137820891740, −6.20611168556265757752315428702, −5.25156913573764715303027534235, −4.96401052213351047208758933447, −3.77363365005615439406795359559, −3.11406349972431004492472570637, −2.36938608187049450453570149939, −1.54148333082103350827083626223, 0,
1.54148333082103350827083626223, 2.36938608187049450453570149939, 3.11406349972431004492472570637, 3.77363365005615439406795359559, 4.96401052213351047208758933447, 5.25156913573764715303027534235, 6.20611168556265757752315428702, 7.22304148114232549137820891740, 7.70375403356815760643800507915
