| L(s) = 1 | + 4-s − 3·9-s + 4·13-s + 16-s − 8·19-s + 25-s − 8·31-s − 3·36-s − 2·37-s + 8·43-s − 14·49-s + 4·52-s + 20·61-s + 64-s − 16·67-s + 20·73-s − 8·76-s − 8·79-s + 9·81-s + 12·97-s + 100-s + 36·109-s − 12·117-s − 6·121-s − 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 9-s + 1.10·13-s + 1/4·16-s − 1.83·19-s + 1/5·25-s − 1.43·31-s − 1/2·36-s − 0.328·37-s + 1.21·43-s − 2·49-s + 0.554·52-s + 2.56·61-s + 1/8·64-s − 1.95·67-s + 2.34·73-s − 0.917·76-s − 0.900·79-s + 81-s + 1.21·97-s + 1/10·100-s + 3.44·109-s − 1.10·117-s − 0.545·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.81059324472633605684417031104, −7.37331018075114938881780514356, −6.87493338522800143592874040603, −6.27790394394954895549345060581, −6.21035074530515276134610094877, −5.72116256637183355409115684013, −5.15564100419176396227847875341, −4.69247680915447547060653501609, −4.00369031444618358374700555311, −3.56437110216151834426597896111, −3.17732882969520207000389302979, −2.26361843784339976106632082372, −2.10151323526117516194563984785, −1.12328850500342707125804122399, 0,
1.12328850500342707125804122399, 2.10151323526117516194563984785, 2.26361843784339976106632082372, 3.17732882969520207000389302979, 3.56437110216151834426597896111, 4.00369031444618358374700555311, 4.69247680915447547060653501609, 5.15564100419176396227847875341, 5.72116256637183355409115684013, 6.21035074530515276134610094877, 6.27790394394954895549345060581, 6.87493338522800143592874040603, 7.37331018075114938881780514356, 7.81059324472633605684417031104
