| L(s) = 1 | − 6·3-s + 21·9-s − 4·11-s + 6·17-s + 2·19-s − 10·25-s − 54·27-s + 24·33-s − 24·41-s − 16·43-s − 13·49-s − 36·51-s − 12·57-s + 10·59-s − 14·67-s + 2·73-s + 60·75-s + 108·81-s − 12·83-s − 8·89-s − 12·97-s − 84·99-s − 10·107-s + 4·113-s − 10·121-s + 144·123-s + 127-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s − 1.20·11-s + 1.45·17-s + 0.458·19-s − 2·25-s − 10.3·27-s + 4.17·33-s − 3.74·41-s − 2.43·43-s − 1.85·49-s − 5.04·51-s − 1.58·57-s + 1.30·59-s − 1.71·67-s + 0.234·73-s + 6.92·75-s + 12·81-s − 1.31·83-s − 0.847·89-s − 1.21·97-s − 8.44·99-s − 0.966·107-s + 0.376·113-s − 0.909·121-s + 12.9·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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
−7.47536575646542499937872009352, −7.34354802745880252390609982428, −6.61303503651954034437075836396, −6.48093639487009967007655018620, −5.91824238738469752095279165720, −5.40675236139494372245760564830, −5.22289133265133441977616849928, −5.05677384969689549249862504577, −4.39011552338323920878309071486, −3.69177683782741766976501946645, −3.11827862407422559284554596552, −1.71114737147421549133826699879, −1.37942708011457790071691730372, 0, 0,
1.37942708011457790071691730372, 1.71114737147421549133826699879, 3.11827862407422559284554596552, 3.69177683782741766976501946645, 4.39011552338323920878309071486, 5.05677384969689549249862504577, 5.22289133265133441977616849928, 5.40675236139494372245760564830, 5.91824238738469752095279165720, 6.48093639487009967007655018620, 6.61303503651954034437075836396, 7.34354802745880252390609982428, 7.47536575646542499937872009352
