| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 14·13-s + 9·19-s + 3·21-s − 5·25-s − 9·27-s + 15·31-s + 37-s − 42·39-s − 6·49-s − 27·57-s − 14·61-s − 6·63-s − 21·67-s − 7·73-s + 15·75-s + 21·79-s + 9·81-s − 14·91-s − 45·93-s − 28·97-s + 33·103-s + 17·109-s − 3·111-s + 84·117-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 3.88·13-s + 2.06·19-s + 0.654·21-s − 25-s − 1.73·27-s + 2.69·31-s + 0.164·37-s − 6.72·39-s − 6/7·49-s − 3.57·57-s − 1.79·61-s − 0.755·63-s − 2.56·67-s − 0.819·73-s + 1.73·75-s + 2.36·79-s + 81-s − 1.46·91-s − 4.66·93-s − 2.84·97-s + 3.25·103-s + 1.62·109-s − 0.284·111-s + 7.76·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107946275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107946275\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.50509623888421216855033106010, −11.42700106021728381473757584862, −11.02778821757714182722028862004, −10.54331343173576626376388242329, −9.992587183552170040469416810071, −9.773730420857100699207285973512, −8.811285232859324080727195548227, −8.742674138928142622161561070810, −7.77938763746407257923605980462, −7.62660824680861555647374364162, −6.48815373607816860859905757546, −6.45631247953826733311180692242, −5.89103341909744038016857788119, −5.74001776421915450440581050507, −4.82736235381145090408970689515, −4.32523645769660684583815987270, −3.54874763146271656919782870647, −3.16191065780191309217603365689, −1.39779143957032138731475895823, −1.02291026116260855523232131320,
1.02291026116260855523232131320, 1.39779143957032138731475895823, 3.16191065780191309217603365689, 3.54874763146271656919782870647, 4.32523645769660684583815987270, 4.82736235381145090408970689515, 5.74001776421915450440581050507, 5.89103341909744038016857788119, 6.45631247953826733311180692242, 6.48815373607816860859905757546, 7.62660824680861555647374364162, 7.77938763746407257923605980462, 8.742674138928142622161561070810, 8.811285232859324080727195548227, 9.773730420857100699207285973512, 9.992587183552170040469416810071, 10.54331343173576626376388242329, 11.02778821757714182722028862004, 11.42700106021728381473757584862, 11.50509623888421216855033106010
