| L(s) = 1 | + 13-s − 4·19-s + 8·25-s + 8·31-s − 5·37-s − 7·43-s − 6·49-s + 7·61-s + 17·67-s + 13·73-s + 11·79-s − 20·97-s + 8·103-s − 14·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.277·13-s − 0.917·19-s + 8/5·25-s + 1.43·31-s − 0.821·37-s − 1.06·43-s − 6/7·49-s + 0.896·61-s + 2.07·67-s + 1.52·73-s + 1.23·79-s − 2.03·97-s + 0.788·103-s − 1.34·109-s − 0.363·121-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 − 0.384·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1306368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1306368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.981705933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981705933\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.188913625445171576195519937768, −7.59681592053697867177660805087, −6.85295844001202131717002507434, −6.69989659357220578890011938424, −6.47294793212139372078492784523, −5.78354266426536567028131828665, −5.20201164250892086313823160076, −4.92572424834588306968535962636, −4.43120811670044730497174339071, −3.79491216224476256414194363061, −3.39560622666269625748553881842, −2.73133017413474813044997205384, −2.23006140132342102247588404627, −1.45803057277367781153766544369, −0.64473397768992702437501751084,
0.64473397768992702437501751084, 1.45803057277367781153766544369, 2.23006140132342102247588404627, 2.73133017413474813044997205384, 3.39560622666269625748553881842, 3.79491216224476256414194363061, 4.43120811670044730497174339071, 4.92572424834588306968535962636, 5.20201164250892086313823160076, 5.78354266426536567028131828665, 6.47294793212139372078492784523, 6.69989659357220578890011938424, 6.85295844001202131717002507434, 7.59681592053697867177660805087, 8.188913625445171576195519937768
