| L(s) = 1 | − 3·9-s + 10·11-s + 12·19-s + 18·29-s + 8·31-s − 8·41-s − 49-s − 16·59-s − 16·61-s − 16·71-s + 26·79-s − 8·89-s − 30·99-s + 12·101-s + 6·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s − 36·171-s + ⋯ |
| L(s) = 1 | − 9-s + 3.01·11-s + 2.75·19-s + 3.34·29-s + 1.43·31-s − 1.24·41-s − 1/7·49-s − 2.08·59-s − 2.04·61-s − 1.89·71-s + 2.92·79-s − 0.847·89-s − 3.01·99-s + 1.19·101-s + 0.574·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s − 2.75·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.871386299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.871386299\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−9.398174719608895208671737881164, −8.723141604878883079882675332289, −8.371968549762333853553724799979, −7.78065667224457195995116578000, −7.60715857593992724692051154627, −6.93309538377953693684717319432, −6.62140291844918863942023604177, −6.42213347772441775879381672322, −5.98924522640846236766681601453, −5.70750324547353301729639771179, −4.92433557260171185480729950246, −4.58725350054203323944738814249, −4.53576490497801236392754977217, −3.54897168076766094487351663615, −3.41204131006809852113083302784, −3.00032808649811991417742536225, −2.55240628993546176703583748691, −1.40040218579437681303907033798, −1.35112956324768647266050146326, −0.73741687274224139332583221697,
0.73741687274224139332583221697, 1.35112956324768647266050146326, 1.40040218579437681303907033798, 2.55240628993546176703583748691, 3.00032808649811991417742536225, 3.41204131006809852113083302784, 3.54897168076766094487351663615, 4.53576490497801236392754977217, 4.58725350054203323944738814249, 4.92433557260171185480729950246, 5.70750324547353301729639771179, 5.98924522640846236766681601453, 6.42213347772441775879381672322, 6.62140291844918863942023604177, 6.93309538377953693684717319432, 7.60715857593992724692051154627, 7.78065667224457195995116578000, 8.371968549762333853553724799979, 8.723141604878883079882675332289, 9.398174719608895208671737881164
