| L(s) = 1 | − 3-s + 7-s + 9-s + 2·19-s − 21-s − 25-s − 27-s − 2·31-s − 2·37-s + 49-s − 2·57-s + 63-s + 75-s + 81-s + 2·93-s − 2·103-s + 2·109-s + 2·111-s + ⋯ |
| L(s) = 1 | − 3-s + 7-s + 9-s + 2·19-s − 21-s − 25-s − 27-s − 2·31-s − 2·37-s + 49-s − 2·57-s + 63-s + 75-s + 81-s + 2·93-s − 2·103-s + 2·109-s + 2·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6707608528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6707608528\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 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
−11.68841272699677495242187028195, −11.04044960332537083325011227439, −10.10347276013047451519086358929, −9.105625617983359030842195493699, −7.75533328304734631711820096907, −7.07777300921746737143774935182, −5.63829724719592511786142975740, −5.08497669148077869683555284781, −3.73742660902658418739249992078, −1.62641474112581694945330454827,
1.62641474112581694945330454827, 3.73742660902658418739249992078, 5.08497669148077869683555284781, 5.63829724719592511786142975740, 7.07777300921746737143774935182, 7.75533328304734631711820096907, 9.105625617983359030842195493699, 10.10347276013047451519086358929, 11.04044960332537083325011227439, 11.68841272699677495242187028195
