| L(s) = 1 | + 3·2-s − 3·3-s + 4·4-s + 3·5-s − 9·6-s + 2·7-s + 3·8-s + 2·9-s + 9·10-s − 3·11-s − 12·12-s + 6·14-s − 9·15-s + 3·16-s + 6·17-s + 6·18-s + 3·19-s + 12·20-s − 6·21-s − 9·22-s − 9·24-s − 2·25-s + 6·27-s + 8·28-s + 3·29-s − 27·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 2·4-s + 1.34·5-s − 3.67·6-s + 0.755·7-s + 1.06·8-s + 2/3·9-s + 2.84·10-s − 0.904·11-s − 3.46·12-s + 1.60·14-s − 2.32·15-s + 3/4·16-s + 1.45·17-s + 1.41·18-s + 0.688·19-s + 2.68·20-s − 1.30·21-s − 1.91·22-s − 1.83·24-s − 2/5·25-s + 1.15·27-s + 1.51·28-s + 0.557·29-s − 4.92·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.015405034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.015405034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 31 T + 423 T^{2} - 31 p T^{3} + p^{2} T^{4} \) |
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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
−10.39469235113878481840110883402, −9.842611353272889075000282340984, −9.159245924366192029806963170055, −8.960171870875045147957662254235, −7.984410843302600345325203557243, −7.65517688832451792158966327856, −7.57891971778310487320048274066, −6.55638763103508474136832820972, −6.10061786304037931337812718176, −6.00701635809046028972676613715, −5.60629874154245458866093246394, −5.38394149739157780591913072623, −4.92170792386454532261323230260, −4.73048112192876361752133074913, −4.10077957951623286033575290346, −3.52684003627174272452693858405, −2.71551082361191645407167010437, −2.55842159840189424469881316217, −1.49503934372324026440084297395, −0.814714983260082074970639749580,
0.814714983260082074970639749580, 1.49503934372324026440084297395, 2.55842159840189424469881316217, 2.71551082361191645407167010437, 3.52684003627174272452693858405, 4.10077957951623286033575290346, 4.73048112192876361752133074913, 4.92170792386454532261323230260, 5.38394149739157780591913072623, 5.60629874154245458866093246394, 6.00701635809046028972676613715, 6.10061786304037931337812718176, 6.55638763103508474136832820972, 7.57891971778310487320048274066, 7.65517688832451792158966327856, 7.984410843302600345325203557243, 8.960171870875045147957662254235, 9.159245924366192029806963170055, 9.842611353272889075000282340984, 10.39469235113878481840110883402
