| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·13-s + 15-s − 2·17-s − 7·19-s − 21-s − 6·23-s + 25-s + 27-s − 8·29-s + 3·31-s − 35-s + 37-s + 2·39-s + 6·41-s + 45-s + 4·47-s − 6·49-s − 2·51-s + 10·53-s − 7·57-s − 4·59-s + 11·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 1.60·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s + 0.538·31-s − 0.169·35-s + 0.164·37-s + 0.320·39-s + 0.937·41-s + 0.149·45-s + 0.583·47-s − 6/7·49-s − 0.280·51-s + 1.37·53-s − 0.927·57-s − 0.520·59-s + 1.40·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.335967320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.335967320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−15.06741815912214, −14.65078799465553, −14.15695654273222, −13.48255712811535, −13.13949127894003, −12.71522306379113, −12.09305132055177, −11.29331702802059, −10.90513346233477, −10.13549107054543, −9.877230541893385, −9.074121062708784, −8.715184078882789, −8.167472042534330, −7.488059204621450, −6.858546324841048, −6.138013098223420, −5.913716727118041, −4.969939886081403, −4.049399231239742, −3.929731090186962, −2.902499735631141, −2.202922155280194, −1.747188480036585, −0.5450451358355510,
0.5450451358355510, 1.747188480036585, 2.202922155280194, 2.902499735631141, 3.929731090186962, 4.049399231239742, 4.969939886081403, 5.913716727118041, 6.138013098223420, 6.858546324841048, 7.488059204621450, 8.167472042534330, 8.715184078882789, 9.074121062708784, 9.877230541893385, 10.13549107054543, 10.90513346233477, 11.29331702802059, 12.09305132055177, 12.71522306379113, 13.13949127894003, 13.48255712811535, 14.15695654273222, 14.65078799465553, 15.06741815912214
