| L(s) = 1 | + 3-s + 2.43·5-s − 1.35·7-s + 9-s − 11-s − 13-s + 2.43·15-s − 3.08·17-s − 2.64·19-s − 1.35·21-s − 1.52·23-s + 0.944·25-s + 27-s + 7.79·29-s − 8.43·31-s − 33-s − 3.29·35-s − 11.0·37-s − 39-s − 8.99·41-s + 6.38·43-s + 2.43·45-s + 6.87·47-s − 5.17·49-s − 3.08·51-s − 8.17·53-s − 2.43·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.09·5-s − 0.510·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.629·15-s − 0.748·17-s − 0.607·19-s − 0.295·21-s − 0.317·23-s + 0.188·25-s + 0.192·27-s + 1.44·29-s − 1.51·31-s − 0.174·33-s − 0.557·35-s − 1.81·37-s − 0.160·39-s − 1.40·41-s + 0.973·43-s + 0.363·45-s + 1.00·47-s − 0.738·49-s − 0.432·51-s − 1.12·53-s − 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 - 6.38T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 - 0.876T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 + 0.703T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 9.08T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−7.58825259490332925325929520320, −6.81798095542072454994265083536, −6.32751247050891941898160214947, −5.50870975630322430707344390105, −4.80951746401864103021255043184, −3.89207934598754429510480156204, −3.03465513724849656581507847026, −2.25060902646150500667262822415, −1.61516875793709502394691575297, 0,
1.61516875793709502394691575297, 2.25060902646150500667262822415, 3.03465513724849656581507847026, 3.89207934598754429510480156204, 4.80951746401864103021255043184, 5.50870975630322430707344390105, 6.32751247050891941898160214947, 6.81798095542072454994265083536, 7.58825259490332925325929520320
