| L(s) = 1 | − 3-s + 5·5-s − 2·9-s + 5·11-s − 5·15-s − 6·23-s + 9·25-s + 5·27-s + 9·31-s − 5·33-s + 10·37-s − 10·45-s + 7·47-s − 49-s + 6·53-s + 25·55-s + 59-s + 17·67-s + 6·69-s + 12·71-s − 9·75-s + 81-s + 4·89-s − 9·93-s + 97-s − 10·99-s + 5·103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.23·5-s − 2/3·9-s + 1.50·11-s − 1.29·15-s − 1.25·23-s + 9/5·25-s + 0.962·27-s + 1.61·31-s − 0.870·33-s + 1.64·37-s − 1.49·45-s + 1.02·47-s − 1/7·49-s + 0.824·53-s + 3.37·55-s + 0.130·59-s + 2.07·67-s + 0.722·69-s + 1.42·71-s − 1.03·75-s + 1/9·81-s + 0.423·89-s − 0.933·93-s + 0.101·97-s − 1.00·99-s + 0.492·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.389622605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.389622605\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−7.73480888406014023234210413923, −6.92337340533267637048939588302, −6.53184924998010210253829038390, −6.32666379660542685814931876941, −5.98534387259835609033847075315, −5.71820998062277801286405785805, −5.20236046696412298363509224494, −4.84054100901307318434265422322, −4.01763616604609581990550998326, −3.91303886412868382417623901349, −2.96260382996438908563244548288, −2.32638354211337545321709972082, −2.19363057236283585729729046560, −1.33092135137838035532706865943, −0.808534847914077609680349963831,
0.808534847914077609680349963831, 1.33092135137838035532706865943, 2.19363057236283585729729046560, 2.32638354211337545321709972082, 2.96260382996438908563244548288, 3.91303886412868382417623901349, 4.01763616604609581990550998326, 4.84054100901307318434265422322, 5.20236046696412298363509224494, 5.71820998062277801286405785805, 5.98534387259835609033847075315, 6.32666379660542685814931876941, 6.53184924998010210253829038390, 6.92337340533267637048939588302, 7.73480888406014023234210413923
