| L(s) = 1 | + 2·5-s + 5·9-s − 4·11-s − 6·13-s + 6·17-s + 12·23-s − 7·25-s + 6·37-s + 10·45-s + 5·49-s − 8·55-s − 20·59-s − 12·65-s + 24·67-s + 16·81-s + 32·83-s + 12·85-s − 20·99-s − 8·103-s + 30·109-s − 12·113-s + 24·115-s − 30·117-s − 10·121-s − 26·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 5/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 2.50·23-s − 7/5·25-s + 0.986·37-s + 1.49·45-s + 5/7·49-s − 1.07·55-s − 2.60·59-s − 1.48·65-s + 2.93·67-s + 16/9·81-s + 3.51·83-s + 1.30·85-s − 2.01·99-s − 0.788·103-s + 2.87·109-s − 1.12·113-s + 2.23·115-s − 2.77·117-s − 0.909·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087569194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087569194\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−11.37168840793787288233172923128, −10.83228872062438302440557434694, −10.32442858055002501858250137540, −10.14138038224859624964234125400, −9.605097811540624682727093877165, −9.430494786937664535759407668439, −8.955636128101187260718184376317, −7.76497461112482200405099189371, −7.74632184836783213897782844514, −7.53689314953457584073246640950, −6.68219372843844639588494615195, −6.41491769231875701474624915748, −5.42281091134998164623675271986, −5.23526122000263984690340009282, −4.81797216331020672173860035645, −4.08821981060230659903242129669, −3.28019333367342851958767060337, −2.57871025879883010529356440435, −1.98700077270846129269061733720, −1.00743657042411183446450197924,
1.00743657042411183446450197924, 1.98700077270846129269061733720, 2.57871025879883010529356440435, 3.28019333367342851958767060337, 4.08821981060230659903242129669, 4.81797216331020672173860035645, 5.23526122000263984690340009282, 5.42281091134998164623675271986, 6.41491769231875701474624915748, 6.68219372843844639588494615195, 7.53689314953457584073246640950, 7.74632184836783213897782844514, 7.76497461112482200405099189371, 8.955636128101187260718184376317, 9.430494786937664535759407668439, 9.605097811540624682727093877165, 10.14138038224859624964234125400, 10.32442858055002501858250137540, 10.83228872062438302440557434694, 11.37168840793787288233172923128
