| L(s) = 1 | + 4-s − 2·5-s − 6·11-s + 16-s − 2·20-s + 3·25-s − 8·31-s + 16·37-s − 6·44-s − 10·49-s + 12·53-s + 12·55-s − 12·59-s + 64-s − 8·67-s + 24·71-s − 2·80-s − 24·89-s + 4·97-s + 3·100-s + 4·103-s − 12·113-s + 25·121-s − 8·124-s − 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 1.80·11-s + 1/4·16-s − 0.447·20-s + 3/5·25-s − 1.43·31-s + 2.63·37-s − 0.904·44-s − 1.42·49-s + 1.64·53-s + 1.61·55-s − 1.56·59-s + 1/8·64-s − 0.977·67-s + 2.84·71-s − 0.223·80-s − 2.54·89-s + 0.406·97-s + 3/10·100-s + 0.394·103-s − 1.12·113-s + 2.27·121-s − 0.718·124-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.133667415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133667415\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−8.137806933221736892131373840887, −7.68212623713153575797344897373, −7.42257774366004622063287945506, −6.90100106970264123787846733101, −6.44043843648522745813285526746, −5.76349049541450034117946945917, −5.57449936126795434832243724178, −4.88210668636964910751844139681, −4.56211949870166416184694202500, −3.88397116255254813881564963884, −3.41872430047913395123890760698, −2.68907161640043859094582658785, −2.49967974392957110591964056962, −1.55969063135541018782660736902, −0.48404849833756435955049600574,
0.48404849833756435955049600574, 1.55969063135541018782660736902, 2.49967974392957110591964056962, 2.68907161640043859094582658785, 3.41872430047913395123890760698, 3.88397116255254813881564963884, 4.56211949870166416184694202500, 4.88210668636964910751844139681, 5.57449936126795434832243724178, 5.76349049541450034117946945917, 6.44043843648522745813285526746, 6.90100106970264123787846733101, 7.42257774366004622063287945506, 7.68212623713153575797344897373, 8.137806933221736892131373840887
