| L(s) = 1 | − 3-s − 2·5-s − 2·9-s + 4·11-s + 2·15-s − 4·23-s + 3·25-s + 5·27-s − 2·31-s − 4·33-s + 4·45-s − 5·49-s − 10·53-s − 8·55-s − 12·59-s − 8·67-s + 4·69-s + 8·71-s − 3·75-s + 81-s − 8·89-s + 2·93-s − 8·97-s − 8·99-s + 8·103-s − 4·113-s + 8·115-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 2/3·9-s + 1.20·11-s + 0.516·15-s − 0.834·23-s + 3/5·25-s + 0.962·27-s − 0.359·31-s − 0.696·33-s + 0.596·45-s − 5/7·49-s − 1.37·53-s − 1.07·55-s − 1.56·59-s − 0.977·67-s + 0.481·69-s + 0.949·71-s − 0.346·75-s + 1/9·81-s − 0.847·89-s + 0.207·93-s − 0.812·97-s − 0.804·99-s + 0.788·103-s − 0.376·113-s + 0.746·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7882179101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7882179101\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.049496042608364298325386541409, −7.23318312754565564074295290655, −7.07122916745680490504336586534, −6.38806944381188641334173315114, −6.23660688601137130397424073835, −5.73116162270533179287506756253, −5.24353532650850934911391864980, −4.59890695124802715126102296732, −4.39809202484285117429348371036, −3.78638220578921326970755932490, −3.28184685371091341154692134872, −2.90742817054475538477715063080, −1.97982848295123695961488886714, −1.35437612691614642348380241004, −0.40310473672986045166249655812,
0.40310473672986045166249655812, 1.35437612691614642348380241004, 1.97982848295123695961488886714, 2.90742817054475538477715063080, 3.28184685371091341154692134872, 3.78638220578921326970755932490, 4.39809202484285117429348371036, 4.59890695124802715126102296732, 5.24353532650850934911391864980, 5.73116162270533179287506756253, 6.23660688601137130397424073835, 6.38806944381188641334173315114, 7.07122916745680490504336586534, 7.23318312754565564074295290655, 8.049496042608364298325386541409
