| L(s) = 1 | − 8·17-s − 16·29-s + 8·37-s − 16·41-s − 8·49-s + 16·53-s − 12·61-s − 8·73-s − 9·81-s + 4·89-s + 8·97-s − 16·101-s + 20·109-s + 32·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.94·17-s − 2.97·29-s + 1.31·37-s − 2.49·41-s − 8/7·49-s + 2.19·53-s − 1.53·61-s − 0.936·73-s − 81-s + 0.423·89-s + 0.812·97-s − 1.59·101-s + 1.91·109-s + 3.01·113-s + 2/11·121-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 − 2·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−7.71832627834747909288774991947, −7.56022923673304581525121920605, −6.98804009558510783854578858897, −6.97929518577251570045348602820, −6.45966312445147218822801253650, −6.07195532910014085295109807669, −5.70104471382508399234376644509, −5.47323264744782359453053984863, −4.91447937823470135637597502559, −4.45454894691241859383511689586, −4.40437736230118030041057856727, −3.79016187240491880953291569848, −3.35717223246557514948188935694, −3.15089600182070909283723960231, −2.26040715267753578253977073249, −2.18628486065288014965636131715, −1.71145543516659841251904292902, −1.07845093473388407883968326075, 0, 0,
1.07845093473388407883968326075, 1.71145543516659841251904292902, 2.18628486065288014965636131715, 2.26040715267753578253977073249, 3.15089600182070909283723960231, 3.35717223246557514948188935694, 3.79016187240491880953291569848, 4.40437736230118030041057856727, 4.45454894691241859383511689586, 4.91447937823470135637597502559, 5.47323264744782359453053984863, 5.70104471382508399234376644509, 6.07195532910014085295109807669, 6.45966312445147218822801253650, 6.97929518577251570045348602820, 6.98804009558510783854578858897, 7.56022923673304581525121920605, 7.71832627834747909288774991947
