| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 2·11-s + 13-s + 3·15-s − 17-s + 19-s + 21-s − 23-s + 4·25-s − 27-s − 9·29-s − 2·31-s − 2·33-s + 3·35-s + 10·37-s − 39-s − 7·41-s − 43-s − 3·45-s + 4·47-s + 49-s + 51-s − 6·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.774·15-s − 0.242·17-s + 0.229·19-s + 0.218·21-s − 0.208·23-s + 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.348·33-s + 0.507·35-s + 1.64·37-s − 0.160·39-s − 1.09·41-s − 0.152·43-s − 0.447·45-s + 0.583·47-s + 1/7·49-s + 0.140·51-s − 0.809·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.025552141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025552141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.71608699076624, −12.03497708555030, −11.76929248511396, −11.31898604151749, −11.14952953913270, −10.46606187656869, −10.01206125272991, −9.418974642291801, −9.072505801042355, −8.495721900535426, −7.954539968794806, −7.575890493216913, −7.070917403848056, −6.687107260133062, −6.103508175522403, −5.632137816950182, −5.098644539257595, −4.430113392908348, −4.003698464171770, −3.654069793637166, −3.160262290828171, −2.324359579400408, −1.688708067985473, −0.8810155453666934, −0.3582700754282636,
0.3582700754282636, 0.8810155453666934, 1.688708067985473, 2.324359579400408, 3.160262290828171, 3.654069793637166, 4.003698464171770, 4.430113392908348, 5.098644539257595, 5.632137816950182, 6.103508175522403, 6.687107260133062, 7.070917403848056, 7.575890493216913, 7.954539968794806, 8.495721900535426, 9.072505801042355, 9.418974642291801, 10.01206125272991, 10.46606187656869, 11.14952953913270, 11.31898604151749, 11.76929248511396, 12.03497708555030, 12.71608699076624
