| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 16-s − 5·19-s + 6·22-s + 8·25-s + 32-s − 5·38-s − 15·41-s + 7·43-s + 6·44-s + 5·49-s + 8·50-s + 3·59-s + 64-s + 10·67-s − 5·73-s − 5·76-s − 15·82-s − 21·83-s + 7·86-s + 6·88-s − 18·89-s + 16·97-s + 5·98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 1/4·16-s − 1.14·19-s + 1.27·22-s + 8/5·25-s + 0.176·32-s − 0.811·38-s − 2.34·41-s + 1.06·43-s + 0.904·44-s + 5/7·49-s + 1.13·50-s + 0.390·59-s + 1/8·64-s + 1.22·67-s − 0.585·73-s − 0.573·76-s − 1.65·82-s − 2.30·83-s + 0.754·86-s + 0.639·88-s − 1.90·89-s + 1.62·97-s + 0.505·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.608369268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.608369268\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.691096920727666192620229970406, −9.010014877757033398103697254493, −8.576197409888677316514387666783, −8.387872955938437774281542278314, −7.32942484318764809603728310383, −6.98238379236555624704380534840, −6.57424985736775969649302612645, −6.09133802938756702920483547738, −5.46838516797411861337239923234, −4.74395453179061788517297811206, −4.27491284360702029443722747717, −3.71412917214443624082862797405, −3.05790357296229359941474219386, −2.13903233770606915583361651786, −1.23567913556734945142120794524,
1.23567913556734945142120794524, 2.13903233770606915583361651786, 3.05790357296229359941474219386, 3.71412917214443624082862797405, 4.27491284360702029443722747717, 4.74395453179061788517297811206, 5.46838516797411861337239923234, 6.09133802938756702920483547738, 6.57424985736775969649302612645, 6.98238379236555624704380534840, 7.32942484318764809603728310383, 8.387872955938437774281542278314, 8.576197409888677316514387666783, 9.010014877757033398103697254493, 9.691096920727666192620229970406
