| L(s) = 1 | − 3-s − 2·4-s − 5-s − 7-s + 9-s + 2·12-s + 5·13-s + 15-s + 4·16-s − 17-s − 2·19-s + 2·20-s + 21-s − 23-s + 25-s − 27-s + 2·28-s − 8·29-s + 31-s + 35-s − 2·36-s − 3·37-s − 5·39-s + 7·41-s − 45-s − 47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.242·17-s − 0.458·19-s + 0.447·20-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.48·29-s + 0.179·31-s + 0.169·35-s − 1/3·36-s − 0.493·37-s − 0.800·39-s + 1.09·41-s − 0.149·45-s − 0.145·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9861657952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9861657952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.94645010367013, −12.67639228355167, −12.13809156131016, −11.51163348914784, −11.07945365248700, −10.78069707811378, −10.21147900761468, −9.623557873621011, −9.265172824128273, −8.780401729730869, −8.252159027044420, −7.909615827579054, −7.263512415392730, −6.702020863456676, −6.114646181239940, −5.821736942247683, −5.195587867198301, −4.687434175309463, −4.116273056397512, −3.623141418337564, −3.422132478011135, −2.389011285923528, −1.665635895944122, −0.9177046462778496, −0.3730164288062393,
0.3730164288062393, 0.9177046462778496, 1.665635895944122, 2.389011285923528, 3.422132478011135, 3.623141418337564, 4.116273056397512, 4.687434175309463, 5.195587867198301, 5.821736942247683, 6.114646181239940, 6.702020863456676, 7.263512415392730, 7.909615827579054, 8.252159027044420, 8.780401729730869, 9.265172824128273, 9.623557873621011, 10.21147900761468, 10.78069707811378, 11.07945365248700, 11.51163348914784, 12.13809156131016, 12.67639228355167, 12.94645010367013
