| L(s) = 1 | + 4-s − 2·5-s − 6·9-s + 4·11-s + 16-s − 2·20-s + 3·25-s + 16·31-s − 6·36-s − 20·37-s + 4·44-s + 12·45-s + 16·47-s + 49-s − 4·53-s − 8·55-s − 16·59-s + 64-s − 24·67-s − 32·71-s − 2·80-s + 27·81-s + 20·89-s + 4·97-s − 24·99-s + 3·100-s + 32·103-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 2·9-s + 1.20·11-s + 1/4·16-s − 0.447·20-s + 3/5·25-s + 2.87·31-s − 36-s − 3.28·37-s + 0.603·44-s + 1.78·45-s + 2.33·47-s + 1/7·49-s − 0.549·53-s − 1.07·55-s − 2.08·59-s + 1/8·64-s − 2.93·67-s − 3.79·71-s − 0.223·80-s + 3·81-s + 2.11·89-s + 0.406·97-s − 2.41·99-s + 3/10·100-s + 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.267889620477558478773970636673, −7.61679473492378752838968818800, −7.45420248729985818641828193491, −6.82056701612064998634186124368, −6.17616144092315208416665412995, −6.12828563996775624727810348890, −5.56548617967314624330824606220, −4.63923341949934829303628265619, −4.61875239739401252269849039001, −3.69516946377050774425439998698, −3.09181792884659532775858950361, −3.00381372509091040158022668795, −2.06078218456002324346666781043, −1.14034834640667941355654881863, 0,
1.14034834640667941355654881863, 2.06078218456002324346666781043, 3.00381372509091040158022668795, 3.09181792884659532775858950361, 3.69516946377050774425439998698, 4.61875239739401252269849039001, 4.63923341949934829303628265619, 5.56548617967314624330824606220, 6.12828563996775624727810348890, 6.17616144092315208416665412995, 6.82056701612064998634186124368, 7.45420248729985818641828193491, 7.61679473492378752838968818800, 8.267889620477558478773970636673
