L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 11-s + 12-s + 13-s + 16-s − 17-s − 22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·31-s + 32-s − 33-s − 34-s + 39-s − 44-s + 2·46-s + 48-s + 50-s − 51-s + 52-s + ⋯ |
L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 11-s + 12-s + 13-s + 16-s − 17-s − 22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·31-s + 32-s − 33-s − 34-s + 39-s − 44-s + 2·46-s + 48-s + 50-s − 51-s + 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.238767036\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.238767036\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−8.802458348725255232615498550120, −8.000561748114039096900652839120, −7.26181370817594147845769564814, −6.59949655405820058488780064255, −5.60066543217778880257595535868, −4.98885668953084857028779377323, −4.01888448090175434953144994679, −3.15952568315949717571176898170, −2.67549620888112336507036147811, −1.60719134774815446283767953176,
1.60719134774815446283767953176, 2.67549620888112336507036147811, 3.15952568315949717571176898170, 4.01888448090175434953144994679, 4.98885668953084857028779377323, 5.60066543217778880257595535868, 6.59949655405820058488780064255, 7.26181370817594147845769564814, 8.000561748114039096900652839120, 8.802458348725255232615498550120