| L(s) = 1 | − 8·11-s + 8·19-s + 12·29-s + 16·31-s + 12·41-s + 14·49-s + 8·59-s − 4·61-s − 16·71-s + 16·79-s − 12·89-s + 36·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 1.83·19-s + 2.22·29-s + 2.87·31-s + 1.87·41-s + 2·49-s + 1.04·59-s − 0.512·61-s − 1.89·71-s + 1.80·79-s − 1.27·89-s + 3.58·101-s + 0.383·109-s + 2.36·121-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480298526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480298526\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.433541222481676647981680445519, −9.166428420982607673566957778252, −8.401721908485674179921045570138, −8.397636889053871211296733033831, −7.78683236001340624141400306782, −7.70510209192506548609191151063, −7.13285487949861029411796248491, −6.83426600679093431030673821281, −6.06117262008612095337677923620, −5.92437372373293846143174271547, −5.40153293476680559978774341125, −4.99511822007522351101088749327, −4.45557799798515875880238002723, −4.43121491682577421235021657710, −3.29469012059653924365266198569, −3.07153291378275590330664718315, −2.48498095307950116295131430390, −2.36290450679035765217036119263, −1.00623222604674244934321727321, −0.74492245495178499138833289746,
0.74492245495178499138833289746, 1.00623222604674244934321727321, 2.36290450679035765217036119263, 2.48498095307950116295131430390, 3.07153291378275590330664718315, 3.29469012059653924365266198569, 4.43121491682577421235021657710, 4.45557799798515875880238002723, 4.99511822007522351101088749327, 5.40153293476680559978774341125, 5.92437372373293846143174271547, 6.06117262008612095337677923620, 6.83426600679093431030673821281, 7.13285487949861029411796248491, 7.70510209192506548609191151063, 7.78683236001340624141400306782, 8.397636889053871211296733033831, 8.401721908485674179921045570138, 9.166428420982607673566957778252, 9.433541222481676647981680445519
