| L(s) = 1 | + 2-s − 2·3-s − 4-s + 5-s − 2·6-s − 3·8-s + 9-s + 10-s + 2·12-s − 3·13-s − 2·15-s − 16-s + 18-s − 20-s + 6·24-s + 25-s − 3·26-s + 4·27-s − 2·30-s − 5·31-s + 5·32-s − 36-s + 21·37-s + 6·39-s − 3·40-s − 12·41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s − 0.832·13-s − 0.516·15-s − 1/4·16-s + 0.235·18-s − 0.223·20-s + 1.22·24-s + 1/5·25-s − 0.588·26-s + 0.769·27-s − 0.365·30-s − 0.898·31-s + 0.883·32-s − 1/6·36-s + 3.45·37-s + 0.960·39-s − 0.474·40-s − 1.87·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.887315493381641812267299163801, −8.374423639377667927524653520276, −7.76808777023899919160207095193, −7.25501711906557838799046022847, −6.67280506232813047140348475762, −6.09074702637691696932449821103, −5.86161561269523103309967568139, −5.31293846655662422468106502640, −4.92577200972167713662162849286, −4.34846347650913340301442118410, −3.92732225976336392665964410174, −2.90913798276067540906908593947, −2.51938651871105125230691073633, −1.20070290550731468156386632464, 0,
1.20070290550731468156386632464, 2.51938651871105125230691073633, 2.90913798276067540906908593947, 3.92732225976336392665964410174, 4.34846347650913340301442118410, 4.92577200972167713662162849286, 5.31293846655662422468106502640, 5.86161561269523103309967568139, 6.09074702637691696932449821103, 6.67280506232813047140348475762, 7.25501711906557838799046022847, 7.76808777023899919160207095193, 8.374423639377667927524653520276, 8.887315493381641812267299163801
