| L(s) = 1 | − 4-s − 8·13-s − 3·16-s + 13·19-s − 4·25-s + 4·31-s − 5·37-s + 10·43-s − 6·49-s + 8·52-s − 8·61-s + 7·64-s + 7·67-s + 16·73-s − 13·76-s − 5·79-s + 7·97-s + 4·100-s − 5·103-s + 16·109-s + 2·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 5·148-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.21·13-s − 3/4·16-s + 2.98·19-s − 4/5·25-s + 0.718·31-s − 0.821·37-s + 1.52·43-s − 6/7·49-s + 1.10·52-s − 1.02·61-s + 7/8·64-s + 0.855·67-s + 1.87·73-s − 1.49·76-s − 0.562·79-s + 0.710·97-s + 2/5·100-s − 0.492·103-s + 1.53·109-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.410·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238707 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238707 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.166037777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166037777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 421 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 35 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.212709185765994872615917055211, −8.516316118849784900317874457219, −7.898956608454661053902671380092, −7.51032126820878630828627184417, −7.25479700968238562483478879688, −6.72884853074259044584197403271, −5.98690415376505432253866730759, −5.36806401679096366777792091708, −5.02072098492052040256329725980, −4.65857275195778184312068288440, −3.93059813538086874936449027672, −3.19153538386113992435277004405, −2.67414174060325907646520992580, −1.89702216041433194956964154284, −0.66035146995117915029773702426,
0.66035146995117915029773702426, 1.89702216041433194956964154284, 2.67414174060325907646520992580, 3.19153538386113992435277004405, 3.93059813538086874936449027672, 4.65857275195778184312068288440, 5.02072098492052040256329725980, 5.36806401679096366777792091708, 5.98690415376505432253866730759, 6.72884853074259044584197403271, 7.25479700968238562483478879688, 7.51032126820878630828627184417, 7.898956608454661053902671380092, 8.516316118849784900317874457219, 9.212709185765994872615917055211
