| L(s) = 1 | + 4·5-s + 5·9-s + 2·11-s − 8·19-s + 11·25-s + 2·29-s + 12·31-s − 20·41-s + 20·45-s − 49-s + 8·55-s + 12·59-s − 8·61-s + 32·71-s − 22·79-s + 16·81-s − 24·89-s − 32·95-s + 10·99-s + 30·109-s − 19·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 5/3·9-s + 0.603·11-s − 1.83·19-s + 11/5·25-s + 0.371·29-s + 2.15·31-s − 3.12·41-s + 2.98·45-s − 1/7·49-s + 1.07·55-s + 1.56·59-s − 1.02·61-s + 3.79·71-s − 2.47·79-s + 16/9·81-s − 2.54·89-s − 3.28·95-s + 1.00·99-s + 2.87·109-s − 1.72·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.113680750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.113680750\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−10.70862297995796741768319571732, −10.32784410326647406069158074081, −9.992745442488012655709516755189, −9.954249134104325285537059379053, −9.380649780749393558087793542342, −8.737230499483633929453071530797, −8.488373296767513634131314970621, −8.029992865074042590792878729040, −7.11430035209296563389491696045, −6.77948403971659315174237405257, −6.40964565176020716426072231885, −6.27082841114926909906127331834, −5.30679125202027908688252562844, −5.01305810317297161823904958957, −4.35998212917175502672056735713, −3.98222919459956599204784297958, −3.06333061034806352466887240392, −2.29131801132705061393555228957, −1.77123885835003897391704340656, −1.16571607928687842361802773531,
1.16571607928687842361802773531, 1.77123885835003897391704340656, 2.29131801132705061393555228957, 3.06333061034806352466887240392, 3.98222919459956599204784297958, 4.35998212917175502672056735713, 5.01305810317297161823904958957, 5.30679125202027908688252562844, 6.27082841114926909906127331834, 6.40964565176020716426072231885, 6.77948403971659315174237405257, 7.11430035209296563389491696045, 8.029992865074042590792878729040, 8.488373296767513634131314970621, 8.737230499483633929453071530797, 9.380649780749393558087793542342, 9.954249134104325285537059379053, 9.992745442488012655709516755189, 10.32784410326647406069158074081, 10.70862297995796741768319571732
