| L(s) = 1 | + 9·13-s + 8·19-s + 5·25-s − 14·31-s − 3·43-s + 11·49-s − 61-s − 11·67-s + 17·73-s + 17·79-s + 24·97-s + 14·103-s + 36·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 41·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.49·13-s + 1.83·19-s + 25-s − 2.51·31-s − 0.457·43-s + 11/7·49-s − 0.128·61-s − 1.34·67-s + 1.98·73-s + 1.91·79-s + 2.43·97-s + 1.37·103-s + 3.44·109-s + 2·121-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 + 3.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.610808825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610808825\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−8.834282809859647600257327435593, −8.826792769283609918924647253538, −8.389837424311508733010995165773, −7.76669957814941154157696719087, −7.42182260520246511530022312767, −7.35512373538926606015075726666, −6.60849502674455476455086979966, −6.38420106638208408962675946852, −5.86507587462275978652565944965, −5.65701979823992623358319537189, −5.10677114382200221371282425923, −4.87351471045695446661129389232, −4.13586770824994939932831582742, −3.68223235472028571593326954320, −3.31518193615009545917030189939, −3.26205863239706769896436969944, −2.25100802577230769197044527166, −1.80689725645942856831485651837, −1.10848548658685434919762646626, −0.73954777372471465594506608956,
0.73954777372471465594506608956, 1.10848548658685434919762646626, 1.80689725645942856831485651837, 2.25100802577230769197044527166, 3.26205863239706769896436969944, 3.31518193615009545917030189939, 3.68223235472028571593326954320, 4.13586770824994939932831582742, 4.87351471045695446661129389232, 5.10677114382200221371282425923, 5.65701979823992623358319537189, 5.86507587462275978652565944965, 6.38420106638208408962675946852, 6.60849502674455476455086979966, 7.35512373538926606015075726666, 7.42182260520246511530022312767, 7.76669957814941154157696719087, 8.389837424311508733010995165773, 8.826792769283609918924647253538, 8.834282809859647600257327435593
