| L(s) = 1 | − 24·13-s + 30·17-s − 40·29-s − 24·37-s + 80·41-s + 49·49-s − 90·53-s + 22·61-s + 96·73-s − 160·89-s + 144·97-s + 40·101-s + 182·109-s − 30·113-s + ⋯ |
| L(s) = 1 | − 1.84·13-s + 1.76·17-s − 1.37·29-s − 0.648·37-s + 1.95·41-s + 49-s − 1.69·53-s + 0.360·61-s + 1.31·73-s − 1.79·89-s + 1.48·97-s + 0.396·101-s + 1.66·109-s − 0.265·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.728721539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728721539\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 24 T + p^{2} T^{2} \) |
| 17 | \( 1 - 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 24 T + p^{2} T^{2} \) |
| 41 | \( 1 - 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 96 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 160 T + p^{2} T^{2} \) |
| 97 | \( 1 - 144 T + p^{2} T^{2} \) |
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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
−8.232668874115872697945408717089, −7.50004260109829634013568091577, −7.20164385906968633512091221714, −5.99708315937770779338119545599, −5.38392211041048996720666229970, −4.64999573175799386035822584243, −3.66137210033531467266415314441, −2.80072238194917238646876666356, −1.87467224570313056305140429990, −0.60985161253589240518429507108,
0.60985161253589240518429507108, 1.87467224570313056305140429990, 2.80072238194917238646876666356, 3.66137210033531467266415314441, 4.64999573175799386035822584243, 5.38392211041048996720666229970, 5.99708315937770779338119545599, 7.20164385906968633512091221714, 7.50004260109829634013568091577, 8.232668874115872697945408717089
