| L(s) = 1 | + 3-s + 9-s + 6·11-s + 3·17-s + 3·19-s + 25-s + 27-s + 6·33-s + 3·41-s + 15·43-s − 2·49-s + 3·51-s + 3·57-s + 12·59-s + 15·67-s + 2·73-s + 75-s + 81-s − 3·89-s − 18·97-s + 6·99-s − 12·107-s + 15·113-s + 6·121-s + 3·123-s + 127-s + 15·129-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.727·17-s + 0.688·19-s + 1/5·25-s + 0.192·27-s + 1.04·33-s + 0.468·41-s + 2.28·43-s − 2/7·49-s + 0.420·51-s + 0.397·57-s + 1.56·59-s + 1.83·67-s + 0.234·73-s + 0.115·75-s + 1/9·81-s − 0.317·89-s − 1.82·97-s + 0.603·99-s − 1.16·107-s + 1.41·113-s + 6/11·121-s + 0.270·123-s + 0.0887·127-s + 1.32·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.943149357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.943149357\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76710401970824300953431778997, −7.33938764215578941891031254506, −6.84750518810796185102905121155, −6.62817409626039038346948461492, −6.08928302188483364198035052919, −5.48679583647861731975404092672, −5.34542395373528490986246079993, −4.42326139898258832900413808050, −4.21155549067905417568092777808, −3.67535959318029793991245816403, −3.33600759982481038544446047763, −2.63085756119108768624560776491, −2.13962865711713201968849350906, −1.26541387152711315677002568416, −0.918514981755483531692757750210,
0.918514981755483531692757750210, 1.26541387152711315677002568416, 2.13962865711713201968849350906, 2.63085756119108768624560776491, 3.33600759982481038544446047763, 3.67535959318029793991245816403, 4.21155549067905417568092777808, 4.42326139898258832900413808050, 5.34542395373528490986246079993, 5.48679583647861731975404092672, 6.08928302188483364198035052919, 6.62817409626039038346948461492, 6.84750518810796185102905121155, 7.33938764215578941891031254506, 7.76710401970824300953431778997
