| L(s) = 1 | − 3-s + 9-s + 2·19-s + 25-s − 27-s + 2·43-s − 49-s − 2·57-s − 2·67-s − 2·73-s − 75-s + 81-s − 2·97-s + ⋯ |
| L(s) = 1 | − 3-s + 9-s + 2·19-s + 25-s − 27-s + 2·43-s − 49-s − 2·57-s − 2·67-s − 2·73-s − 75-s + 81-s − 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7523588165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7523588165\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−10.61481630022839060559538574268, −9.788375506036270734574360197277, −9.012122620528699950514954261086, −7.69157613434863384180425944828, −7.07069176772492026409901474511, −6.01155221255283003727964277162, −5.26068862264555903975388920270, −4.33421242306601584657258446493, −3.02202024484294469444376997779, −1.24462410225382491238883086831,
1.24462410225382491238883086831, 3.02202024484294469444376997779, 4.33421242306601584657258446493, 5.26068862264555903975388920270, 6.01155221255283003727964277162, 7.07069176772492026409901474511, 7.69157613434863384180425944828, 9.012122620528699950514954261086, 9.788375506036270734574360197277, 10.61481630022839060559538574268
