| L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·13-s − 4·19-s + 4·21-s + 6·25-s − 27-s − 20·31-s + 20·37-s − 2·39-s + 8·43-s − 2·49-s + 4·57-s − 28·61-s − 4·63-s + 4·67-s − 20·73-s − 6·75-s − 32·79-s + 81-s − 8·91-s + 20·93-s − 4·97-s − 16·103-s − 4·109-s − 20·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.917·19-s + 0.872·21-s + 6/5·25-s − 0.192·27-s − 3.59·31-s + 3.28·37-s − 0.320·39-s + 1.21·43-s − 2/7·49-s + 0.529·57-s − 3.58·61-s − 0.503·63-s + 0.488·67-s − 2.34·73-s − 0.692·75-s − 3.60·79-s + 1/9·81-s − 0.838·91-s + 2.07·93-s − 0.406·97-s − 1.57·103-s − 0.383·109-s − 1.89·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.399166260213490339159845776420, −9.272613788586316089970782456314, −8.740006195634140431482743685580, −7.965645572263157998665195857010, −7.33786979345918290465240439010, −7.00743292844584939053022110799, −6.30352193131009441178367980896, −5.89249065254499898615542236081, −5.63885615575172245866908586348, −4.45984308910786813852795903758, −4.22765636287388069600769197082, −3.27214076659366118813611445645, −2.78430186696517884115720808997, −1.53061467325951463049581429920, 0,
1.53061467325951463049581429920, 2.78430186696517884115720808997, 3.27214076659366118813611445645, 4.22765636287388069600769197082, 4.45984308910786813852795903758, 5.63885615575172245866908586348, 5.89249065254499898615542236081, 6.30352193131009441178367980896, 7.00743292844584939053022110799, 7.33786979345918290465240439010, 7.965645572263157998665195857010, 8.740006195634140431482743685580, 9.272613788586316089970782456314, 9.399166260213490339159845776420
