| L(s) = 1 | − 32·17-s + 14·25-s − 160·41-s + 98·49-s − 220·73-s − 320·89-s + 260·97-s − 448·113-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.88·17-s + 0.559·25-s − 3.90·41-s + 2·49-s − 3.01·73-s − 3.59·89-s + 2.68·97-s − 3.96·113-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6926737245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6926737245\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )( 1 + 8 T + p^{2} T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )( 1 + 40 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )^{2} \) |
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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
−10.03861526867655344386971230962, −9.254199709009993512700537041632, −8.902764851576151584650721114479, −8.596013045298357662910777678869, −8.433430113256166641338464258535, −7.67785907998122309954182002824, −7.30656248423456147505091466522, −6.71436543528194364279836376327, −6.70475617947252912149667465294, −6.11397136595347890989645876625, −5.47919916951654066290406279259, −5.09349811819188437574822220642, −4.70379665261928080575722120619, −3.96162634010145383961794595261, −3.90369058146725972242021472761, −2.83502545400421208775575713893, −2.73815613942736171087425113473, −1.81282405276254602568552654642, −1.40452223650058313724622699356, −0.24549695887320798743080203767,
0.24549695887320798743080203767, 1.40452223650058313724622699356, 1.81282405276254602568552654642, 2.73815613942736171087425113473, 2.83502545400421208775575713893, 3.90369058146725972242021472761, 3.96162634010145383961794595261, 4.70379665261928080575722120619, 5.09349811819188437574822220642, 5.47919916951654066290406279259, 6.11397136595347890989645876625, 6.70475617947252912149667465294, 6.71436543528194364279836376327, 7.30656248423456147505091466522, 7.67785907998122309954182002824, 8.433430113256166641338464258535, 8.596013045298357662910777678869, 8.902764851576151584650721114479, 9.254199709009993512700537041632, 10.03861526867655344386971230962
