L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 6·13-s + 15-s + 2·17-s − 19-s − 4·21-s − 8·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·35-s + 10·37-s − 6·39-s + 6·41-s + 8·43-s + 45-s + 9·49-s + 2·51-s − 2·53-s − 57-s + 12·59-s − 2·61-s − 4·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.66·13-s + 0.258·15-s + 0.485·17-s − 0.229·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s − 0.960·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s − 0.274·53-s − 0.132·57-s + 1.56·59-s − 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.802197573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.802197573\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.194199987425558449696709463708, −7.71070006735544405781278264501, −6.78742746941597310070946594716, −6.27911827133817578485713245751, −5.48856454668411663868014901535, −4.45694864710117237357181717963, −3.73029206579143368766083893310, −2.61936159640480811856196689003, −2.39484585127316782428225546142, −0.69506544771832586053069136501,
0.69506544771832586053069136501, 2.39484585127316782428225546142, 2.61936159640480811856196689003, 3.73029206579143368766083893310, 4.45694864710117237357181717963, 5.48856454668411663868014901535, 6.27911827133817578485713245751, 6.78742746941597310070946594716, 7.71070006735544405781278264501, 8.194199987425558449696709463708