| L(s) = 1 | + 3-s + 4-s + 7-s + 12-s + 16-s − 2·17-s − 19-s + 21-s − 27-s + 28-s − 29-s − 41-s + 48-s − 2·51-s + 53-s − 57-s + 59-s + 64-s − 2·68-s + 2·71-s − 76-s − 79-s − 81-s + 84-s − 87-s + 107-s − 108-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 7-s + 12-s + 16-s − 2·17-s − 19-s + 21-s − 27-s + 28-s − 29-s − 41-s + 48-s − 2·51-s + 53-s − 57-s + 59-s + 64-s − 2·68-s + 2·71-s − 76-s − 79-s − 81-s + 84-s − 87-s + 107-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.874907245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874907245\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.575552668289093712507291927588, −8.569072906770956331703562258530, −8.327361806126993696233944786936, −7.32157799238203900178086682084, −6.64902770196150862581847474014, −5.64460632545848170711734135636, −4.52378546512470601090537939169, −3.55218242726383958414665691500, −2.32829132030591627749093925231, −1.93736752655572495720046345560,
1.93736752655572495720046345560, 2.32829132030591627749093925231, 3.55218242726383958414665691500, 4.52378546512470601090537939169, 5.64460632545848170711734135636, 6.64902770196150862581847474014, 7.32157799238203900178086682084, 8.327361806126993696233944786936, 8.569072906770956331703562258530, 9.575552668289093712507291927588
