L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 5·11-s − 14-s + 16-s + 2·17-s − 19-s + 20-s + 5·22-s − 23-s − 4·25-s − 28-s + 4·29-s − 9·31-s + 32-s + 2·34-s − 35-s + 5·37-s − 38-s + 40-s − 9·41-s − 10·43-s + 5·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.208·23-s − 4/5·25-s − 0.188·28-s + 0.742·29-s − 1.61·31-s + 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.821·37-s − 0.162·38-s + 0.158·40-s − 1.40·41-s − 1.52·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.211658448\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.211658448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−11.66207448326126921977789851208, −10.47585711653745414595121031437, −9.614675318763964452394438121737, −8.693710276630941434637211750501, −7.33295252843734147038723504601, −6.41477474003074322919370438778, −5.64016362400921902023132431260, −4.30159478875183316412804698377, −3.30945073751881784724147936210, −1.71709282690249260530994735777,
1.71709282690249260530994735777, 3.30945073751881784724147936210, 4.30159478875183316412804698377, 5.64016362400921902023132431260, 6.41477474003074322919370438778, 7.33295252843734147038723504601, 8.693710276630941434637211750501, 9.614675318763964452394438121737, 10.47585711653745414595121031437, 11.66207448326126921977789851208