L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s + 9-s + 4·10-s + 6·13-s + 16-s + 18-s + 4·20-s + 11·25-s + 6·26-s − 12·29-s + 32-s + 36-s + 4·37-s + 4·40-s + 6·41-s + 4·45-s − 10·49-s + 11·50-s + 6·52-s − 8·53-s − 12·58-s + 10·61-s + 64-s + 24·65-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.66·13-s + 1/4·16-s + 0.235·18-s + 0.894·20-s + 11/5·25-s + 1.17·26-s − 2.22·29-s + 0.176·32-s + 1/6·36-s + 0.657·37-s + 0.632·40-s + 0.937·41-s + 0.596·45-s − 1.42·49-s + 1.55·50-s + 0.832·52-s − 1.09·53-s − 1.57·58-s + 1.28·61-s + 1/8·64-s + 2.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.918157938\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.918157938\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.059180427820411946921324177435, −7.42827033377366211335095263180, −7.04493519393215511287120775037, −6.35521447322490957495695835547, −6.30892570052987143527596463549, −5.77444425493339557271004596461, −5.52519069981901535713718361121, −4.94676997495496599981061786248, −4.49908851101798957387352897656, −3.72294001178398820850947941384, −3.52524965727212619890826313737, −2.75593113438994678915800945306, −2.10683061542955609070738859573, −1.68982922865917915048180301585, −1.04787078796985986331305125370,
1.04787078796985986331305125370, 1.68982922865917915048180301585, 2.10683061542955609070738859573, 2.75593113438994678915800945306, 3.52524965727212619890826313737, 3.72294001178398820850947941384, 4.49908851101798957387352897656, 4.94676997495496599981061786248, 5.52519069981901535713718361121, 5.77444425493339557271004596461, 6.30892570052987143527596463549, 6.35521447322490957495695835547, 7.04493519393215511287120775037, 7.42827033377366211335095263180, 8.059180427820411946921324177435