| L(s) = 1 | − 1.41·3-s − 1.41·5-s − 0.999·9-s + 11-s + 2.00·15-s + 5.65·17-s − 6·23-s − 2.99·25-s + 5.65·27-s − 6·29-s + 7.07·31-s − 1.41·33-s + 6·37-s − 8·43-s + 1.41·45-s + 1.41·47-s − 8.00·51-s − 1.41·55-s + 9.89·59-s + 8.48·61-s + 14·67-s + 8.48·69-s + 2·71-s − 2.82·73-s + 4.24·75-s − 5.00·81-s − 5.65·83-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.632·5-s − 0.333·9-s + 0.301·11-s + 0.516·15-s + 1.37·17-s − 1.25·23-s − 0.599·25-s + 1.08·27-s − 1.11·29-s + 1.27·31-s − 0.246·33-s + 0.986·37-s − 1.21·43-s + 0.210·45-s + 0.206·47-s − 1.12·51-s − 0.190·55-s + 1.28·59-s + 1.08·61-s + 1.71·67-s + 1.02·69-s + 0.237·71-s − 0.331·73-s + 0.489·75-s − 0.555·81-s − 0.620·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.123952390008028351281560859409, −7.27584587341239895467186444186, −6.45318809596552509218704668193, −5.75269990617007571467183872697, −5.22064788458447409928095265405, −4.16753807397350904508868732486, −3.56374949663012936387291295695, −2.47113582716450103853613014791, −1.13927850455679922256749029927, 0,
1.13927850455679922256749029927, 2.47113582716450103853613014791, 3.56374949663012936387291295695, 4.16753807397350904508868732486, 5.22064788458447409928095265405, 5.75269990617007571467183872697, 6.45318809596552509218704668193, 7.27584587341239895467186444186, 8.123952390008028351281560859409
