| L(s) = 1 | − 4-s − 2·5-s − 10·11-s + 16-s + 12·19-s + 2·20-s − 25-s + 18·29-s − 10·31-s − 4·41-s + 10·44-s − 2·49-s + 20·55-s + 8·59-s + 16·61-s − 64-s − 12·71-s − 12·76-s − 18·79-s − 2·80-s + 28·89-s − 24·95-s + 100-s + 34·101-s + 16·109-s − 18·116-s + 53·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 3.01·11-s + 1/4·16-s + 2.75·19-s + 0.447·20-s − 1/5·25-s + 3.34·29-s − 1.79·31-s − 0.624·41-s + 1.50·44-s − 2/7·49-s + 2.69·55-s + 1.04·59-s + 2.04·61-s − 1/8·64-s − 1.42·71-s − 1.37·76-s − 2.02·79-s − 0.223·80-s + 2.96·89-s − 2.46·95-s + 1/10·100-s + 3.38·101-s + 1.53·109-s − 1.67·116-s + 4.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7849239150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7849239150\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.89769067374391636887529350098, −11.87561603245051575796268994178, −11.31793659471600044250927206799, −10.53776100633995363109203415160, −10.30254307160049165617256383204, −9.969847939962603224254747069476, −9.378758793474960959401271820450, −8.513342707833681403545477686699, −8.338004551349270363214783236731, −7.79769039496498221610434142073, −7.32617079753435338208266015323, −7.10935009136792869828471020355, −5.90046417594827532212043252229, −5.48703993365752463016734000052, −4.86668218986048027350093842957, −4.71365690053750460241576598686, −3.44381070184137633120110012734, −3.17463155682276697949704959154, −2.36924923277547711882102037840, −0.69160204346165612079890437469,
0.69160204346165612079890437469, 2.36924923277547711882102037840, 3.17463155682276697949704959154, 3.44381070184137633120110012734, 4.71365690053750460241576598686, 4.86668218986048027350093842957, 5.48703993365752463016734000052, 5.90046417594827532212043252229, 7.10935009136792869828471020355, 7.32617079753435338208266015323, 7.79769039496498221610434142073, 8.338004551349270363214783236731, 8.513342707833681403545477686699, 9.378758793474960959401271820450, 9.969847939962603224254747069476, 10.30254307160049165617256383204, 10.53776100633995363109203415160, 11.31793659471600044250927206799, 11.87561603245051575796268994178, 11.89769067374391636887529350098
