| L(s) = 1 | − 1.26·3-s − 1.39·9-s + 6.43·11-s + 1.10·13-s + 4.50·17-s + 5.87·19-s + 8.77·23-s + 5.56·27-s + 3.87·29-s + 6.66·31-s − 8.13·33-s − 1.42·37-s − 1.39·39-s + 11.8·41-s − 6.46·43-s + 8.46·47-s − 5.69·51-s + 0.768·53-s − 7.43·57-s + 1.79·59-s − 3.03·61-s − 2.85·67-s − 11.1·69-s − 9.47·71-s − 13.3·73-s − 9.30·79-s − 2.84·81-s + ⋯ |
| L(s) = 1 | − 0.730·3-s − 0.466·9-s + 1.93·11-s + 0.306·13-s + 1.09·17-s + 1.34·19-s + 1.82·23-s + 1.07·27-s + 0.719·29-s + 1.19·31-s − 1.41·33-s − 0.234·37-s − 0.223·39-s + 1.84·41-s − 0.986·43-s + 1.23·47-s − 0.797·51-s + 0.105·53-s − 0.984·57-s + 0.233·59-s − 0.389·61-s − 0.348·67-s − 1.33·69-s − 1.12·71-s − 1.56·73-s − 1.04·79-s − 0.316·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317241862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317241862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.26T + 3T^{2} \) |
| 11 | \( 1 - 6.43T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 8.77T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 - 0.768T + 53T^{2} \) |
| 59 | \( 1 - 1.79T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 97 | \( 1 + 5.96T + 97T^{2} \) |
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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
−7.44311482765023392697553534963, −6.96624057022755079220688923335, −6.17367990866457416476483444110, −5.77955402401601250953073506613, −4.96710084098004061241340599514, −4.29511015372575011978777344474, −3.34205049204850152267675928229, −2.82811962175565926500938688949, −1.19760042020697164609237846162, −0.972120263588630821892113890669,
0.972120263588630821892113890669, 1.19760042020697164609237846162, 2.82811962175565926500938688949, 3.34205049204850152267675928229, 4.29511015372575011978777344474, 4.96710084098004061241340599514, 5.77955402401601250953073506613, 6.17367990866457416476483444110, 6.96624057022755079220688923335, 7.44311482765023392697553534963
