| L(s) = 1 | − 1.76·3-s + 2.55·5-s + 0.100·9-s + 4.25·11-s − 0.715·13-s − 4.50·15-s − 1.47·17-s + 19-s − 5.33·23-s + 1.54·25-s + 5.10·27-s + 2.38·29-s − 0.259·31-s − 7.49·33-s − 3.90·37-s + 1.26·39-s + 5.32·41-s + 0.379·43-s + 0.256·45-s + 13.5·47-s + 2.59·51-s − 4.69·53-s + 10.8·55-s − 1.76·57-s − 7.73·59-s − 2.13·61-s − 1.83·65-s + ⋯ |
| L(s) = 1 | − 1.01·3-s + 1.14·5-s + 0.0334·9-s + 1.28·11-s − 0.198·13-s − 1.16·15-s − 0.357·17-s + 0.229·19-s − 1.11·23-s + 0.308·25-s + 0.982·27-s + 0.442·29-s − 0.0465·31-s − 1.30·33-s − 0.642·37-s + 0.201·39-s + 0.831·41-s + 0.0578·43-s + 0.0382·45-s + 1.97·47-s + 0.363·51-s − 0.644·53-s + 1.46·55-s − 0.233·57-s − 1.00·59-s − 0.273·61-s − 0.227·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.708182189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708182189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 2.55T + 5T^{2} \) |
| 11 | \( 1 - 4.25T + 11T^{2} \) |
| 13 | \( 1 + 0.715T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.259T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 - 5.32T + 41T^{2} \) |
| 43 | \( 1 - 0.379T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 + 2.13T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 - 4.41T + 79T^{2} \) |
| 83 | \( 1 + 0.547T + 83T^{2} \) |
| 89 | \( 1 - 0.672T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.80027605638358286748556007908, −6.92998695432718607013865111743, −6.23237947562546831598922009693, −5.99641850407246342590497670868, −5.24156598055003580505362632932, −4.49129637092278342192937371259, −3.65856721002275255385536408394, −2.50775559144956803587605270492, −1.70463816095883219297258525276, −0.70820038070232299824455572113,
0.70820038070232299824455572113, 1.70463816095883219297258525276, 2.50775559144956803587605270492, 3.65856721002275255385536408394, 4.49129637092278342192937371259, 5.24156598055003580505362632932, 5.99641850407246342590497670868, 6.23237947562546831598922009693, 6.92998695432718607013865111743, 7.80027605638358286748556007908
