| L(s) = 1 | − 7-s + 7·19-s − 5·25-s − 7·31-s − 37-s − 6·49-s + 12·61-s + 21·67-s + 27·73-s − 21·79-s − 7·103-s − 17·109-s − 11·121-s + 127-s + 131-s − 7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 5·175-s + 179-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.60·19-s − 25-s − 1.25·31-s − 0.164·37-s − 6/7·49-s + 1.53·61-s + 2.56·67-s + 3.16·73-s − 2.36·79-s − 0.689·103-s − 1.62·109-s − 121-s + 0.0887·127-s + 0.0873·131-s − 0.606·133-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 + 1.76·169-s + 0.0760·173-s + 0.377·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.692415891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692415891\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.965929396404426893290496771668, −9.739803234288223043280955133882, −9.451347945434611919375696108186, −9.075209716678366122404861401926, −8.454709756491545254317604032990, −8.003099443596802996394711852478, −7.79864188383429250919905254276, −7.18883460906348679797898720501, −6.74514899492623733820970492693, −6.55106597839363908375567308395, −5.72106184272731018574500141570, −5.38150801218405136999067529400, −5.21783925980191247669113442937, −4.38804344520137979405502594660, −3.78116771785722562401599235867, −3.53719274858597482009134066226, −2.87612164784470741835849573784, −2.22046419765715667505445999289, −1.55064695288380308160382172238, −0.60975854407603169031591438648,
0.60975854407603169031591438648, 1.55064695288380308160382172238, 2.22046419765715667505445999289, 2.87612164784470741835849573784, 3.53719274858597482009134066226, 3.78116771785722562401599235867, 4.38804344520137979405502594660, 5.21783925980191247669113442937, 5.38150801218405136999067529400, 5.72106184272731018574500141570, 6.55106597839363908375567308395, 6.74514899492623733820970492693, 7.18883460906348679797898720501, 7.79864188383429250919905254276, 8.003099443596802996394711852478, 8.454709756491545254317604032990, 9.075209716678366122404861401926, 9.451347945434611919375696108186, 9.739803234288223043280955133882, 9.965929396404426893290496771668
