| L(s) = 1 | + 15·9-s + 32·13-s − 6·17-s + 24·29-s + 100·37-s − 126·41-s + 86·49-s + 36·53-s + 52·61-s − 34·73-s + 144·81-s + 198·89-s − 268·97-s + 300·101-s − 148·109-s − 402·113-s + 480·117-s + 95·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 90·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 2.46·13-s − 0.352·17-s + 0.827·29-s + 2.70·37-s − 3.07·41-s + 1.75·49-s + 0.679·53-s + 0.852·61-s − 0.465·73-s + 16/9·81-s + 2.22·89-s − 2.76·97-s + 2.97·101-s − 1.35·109-s − 3.55·113-s + 4.10·117-s + 0.785·121-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 − 0.588·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.362554031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.362554031\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 p T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 86 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 95 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 215 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 970 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 950 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 63 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4226 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7895 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9314 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 4982 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4031 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 99 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 134 T + p^{2} T^{2} )^{2} \) |
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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
−11.22250945086472859096230746346, −10.78208270285007655701555071223, −10.26924741125960761216972452601, −10.17218994527979321910869449092, −9.435200731740931458701078426523, −8.992759792124278948220260051292, −8.546073389937668165749023526460, −8.079835516223184665033781432052, −7.64463526747714171654135654979, −6.90987205146767550308627701424, −6.58555205902764857029126332483, −6.21407177804606843405909681776, −5.53630853151705860169392543435, −4.90128128090932773692779102635, −4.11843238975258540141737791842, −3.99701680205595770402595820942, −3.27444446631699512803414362194, −2.33236338086504263239008724722, −1.41584355203007226398978741247, −0.943918874940286100881959001616,
0.943918874940286100881959001616, 1.41584355203007226398978741247, 2.33236338086504263239008724722, 3.27444446631699512803414362194, 3.99701680205595770402595820942, 4.11843238975258540141737791842, 4.90128128090932773692779102635, 5.53630853151705860169392543435, 6.21407177804606843405909681776, 6.58555205902764857029126332483, 6.90987205146767550308627701424, 7.64463526747714171654135654979, 8.079835516223184665033781432052, 8.546073389937668165749023526460, 8.992759792124278948220260051292, 9.435200731740931458701078426523, 10.17218994527979321910869449092, 10.26924741125960761216972452601, 10.78208270285007655701555071223, 11.22250945086472859096230746346
