| L(s) = 1 | − 2.65·2-s + 2.39·3-s + 5.05·4-s + 3.65·5-s − 6.36·6-s − 8.10·8-s + 2.74·9-s − 9.70·10-s + 0.655·11-s + 12.1·12-s − 13-s + 8.75·15-s + 11.4·16-s + 2.39·17-s − 7.27·18-s − 2.70·19-s + 18.4·20-s − 1.74·22-s + 7.36·23-s − 19.4·24-s + 8.36·25-s + 2.65·26-s − 0.621·27-s − 0.208·29-s − 23.2·30-s − 1.13·31-s − 14.1·32-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 1.38·3-s + 2.52·4-s + 1.63·5-s − 2.59·6-s − 2.86·8-s + 0.913·9-s − 3.06·10-s + 0.197·11-s + 3.49·12-s − 0.277·13-s + 2.26·15-s + 2.85·16-s + 0.581·17-s − 1.71·18-s − 0.620·19-s + 4.12·20-s − 0.371·22-s + 1.53·23-s − 3.96·24-s + 1.67·25-s + 0.520·26-s − 0.119·27-s − 0.0386·29-s − 4.24·30-s − 0.204·31-s − 2.49·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367992418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367992418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 11 | \( 1 - 0.655T + 11T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 + 0.208T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 5.25T + 53T^{2} \) |
| 59 | \( 1 - 8.25T + 59T^{2} \) |
| 61 | \( 1 + 1.89T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 6.75T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 1.51T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−10.14116832724985271347958535562, −9.529867127642993661590570600186, −8.889183321749360410690022288154, −8.431020605800283352557996123618, −7.29079128890212255960207671977, −6.62127784116218580160991895086, −5.43142123789935159051410651735, −3.18549583824662157349818568266, −2.28941462341848929539677406425, −1.47501913933754980833286348448,
1.47501913933754980833286348448, 2.28941462341848929539677406425, 3.18549583824662157349818568266, 5.43142123789935159051410651735, 6.62127784116218580160991895086, 7.29079128890212255960207671977, 8.431020605800283352557996123618, 8.889183321749360410690022288154, 9.529867127642993661590570600186, 10.14116832724985271347958535562
