| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 5·11-s − 5·13-s − 15-s + 17-s + 2·19-s + 2·21-s − 23-s + 25-s + 27-s + 8·29-s + 7·31-s − 5·33-s − 2·35-s − 4·37-s − 5·39-s + 9·41-s − 43-s − 45-s − 8·47-s − 3·49-s + 51-s − 5·53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 1.38·13-s − 0.258·15-s + 0.242·17-s + 0.458·19-s + 0.436·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s + 1.25·31-s − 0.870·33-s − 0.338·35-s − 0.657·37-s − 0.800·39-s + 1.40·41-s − 0.152·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.140·51-s − 0.686·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022273374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022273374\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−16.37590311912186, −15.95097773141617, −15.43879294654153, −14.87968805580868, −14.27357322660412, −13.92661685831113, −13.10673612041180, −12.57388615466122, −11.99927943307370, −11.43907567824869, −10.65939597482300, −10.06514639672663, −9.707400191504651, −8.687282389067547, −8.139624665171793, −7.741612446173234, −7.238357639242382, −6.372693668755784, −5.333940388365975, −4.826709069915779, −4.357712838251003, −3.122351033718072, −2.719915845860538, −1.865483258453076, −0.6323375531482491,
0.6323375531482491, 1.865483258453076, 2.719915845860538, 3.122351033718072, 4.357712838251003, 4.826709069915779, 5.333940388365975, 6.372693668755784, 7.238357639242382, 7.741612446173234, 8.139624665171793, 8.687282389067547, 9.707400191504651, 10.06514639672663, 10.65939597482300, 11.43907567824869, 11.99927943307370, 12.57388615466122, 13.10673612041180, 13.92661685831113, 14.27357322660412, 14.87968805580868, 15.43879294654153, 15.95097773141617, 16.37590311912186
