| L(s) = 1 | + 2·3-s − 5-s + 9-s + 2·11-s − 13-s − 2·15-s + 2·17-s + 2·19-s − 2·23-s + 25-s − 4·27-s + 6·29-s − 2·31-s + 4·33-s + 6·37-s − 2·39-s + 2·41-s + 6·43-s − 45-s + 8·47-s − 7·49-s + 4·51-s + 2·53-s − 2·55-s + 4·57-s + 6·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.359·31-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.560·51-s + 0.274·53-s − 0.269·55-s + 0.529·57-s + 0.781·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.765296059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765296059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.411115000826089753996710903934, −7.76319041513224922753287966748, −7.22602053082602449735325415510, −6.28727870819603758957301866812, −5.43479364046733402754835461064, −4.38796175153128331239805240266, −3.73594560911583979412114290925, −2.95784688889706959154847700703, −2.18201972761241363090254415749, −0.910977372178463170757222920731,
0.910977372178463170757222920731, 2.18201972761241363090254415749, 2.95784688889706959154847700703, 3.73594560911583979412114290925, 4.38796175153128331239805240266, 5.43479364046733402754835461064, 6.28727870819603758957301866812, 7.22602053082602449735325415510, 7.76319041513224922753287966748, 8.411115000826089753996710903934
