L(s) = 1 | + 2·5-s + 9-s − 2·11-s + 10·13-s + 12·17-s + 6·19-s + 2·23-s + 2·29-s + 8·31-s + 2·37-s + 6·41-s + 8·43-s + 2·45-s − 16·47-s − 2·53-s − 4·55-s − 4·59-s + 18·61-s + 20·65-s + 8·67-s − 14·71-s + 6·73-s − 8·81-s − 16·83-s + 24·85-s − 8·89-s + 12·95-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1/3·9-s − 0.603·11-s + 2.77·13-s + 2.91·17-s + 1.37·19-s + 0.417·23-s + 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.298·45-s − 2.33·47-s − 0.274·53-s − 0.539·55-s − 0.520·59-s + 2.30·61-s + 2.48·65-s + 0.977·67-s − 1.66·71-s + 0.702·73-s − 8/9·81-s − 1.75·83-s + 2.60·85-s − 0.847·89-s + 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.575287268\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.575287268\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 100 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 115 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.977690224197909041124311325423, −7.904806215933294428776751238964, −7.16463159935505704374122427892, −6.96247438796743979447106023372, −6.39219917329537215989095654990, −6.25589967543361127439518887901, −5.70612710454375339430707773263, −5.69003880685543450766270264325, −5.20638565994174340742188263959, −5.13104302106102956406358208994, −4.28923402161824780917008581609, −4.01105118044731534360644395993, −3.68798666995465885207399270648, −3.16732717486347104028602724882, −2.80003698277562538141311438473, −2.77584002743352011984606403540, −1.71714779395425520239493176317, −1.31533137676839461678616572811, −1.18440293302880180007687105505, −0.73560815900512632775081729613,
0.73560815900512632775081729613, 1.18440293302880180007687105505, 1.31533137676839461678616572811, 1.71714779395425520239493176317, 2.77584002743352011984606403540, 2.80003698277562538141311438473, 3.16732717486347104028602724882, 3.68798666995465885207399270648, 4.01105118044731534360644395993, 4.28923402161824780917008581609, 5.13104302106102956406358208994, 5.20638565994174340742188263959, 5.69003880685543450766270264325, 5.70612710454375339430707773263, 6.25589967543361127439518887901, 6.39219917329537215989095654990, 6.96247438796743979447106023372, 7.16463159935505704374122427892, 7.904806215933294428776751238964, 7.977690224197909041124311325423