| L(s) = 1 | − 2·5-s − 2·11-s + 2·13-s + 4·17-s + 6·19-s + 2·25-s + 6·29-s + 16·31-s − 6·37-s + 10·43-s + 16·47-s + 10·49-s − 10·53-s + 4·55-s + 6·59-s + 18·61-s − 4·65-s − 10·67-s + 2·83-s − 8·85-s − 12·95-s − 4·97-s + 22·101-s + 14·107-s − 6·109-s + 12·113-s + 2·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s + 2/5·25-s + 1.11·29-s + 2.87·31-s − 0.986·37-s + 1.52·43-s + 2.33·47-s + 10/7·49-s − 1.37·53-s + 0.539·55-s + 0.781·59-s + 2.30·61-s − 0.496·65-s − 1.22·67-s + 0.219·83-s − 0.867·85-s − 1.23·95-s − 0.406·97-s + 2.18·101-s + 1.35·107-s − 0.574·109-s + 1.12·113-s + 2/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236819119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236819119\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.18467944199865474120508380945, −9.658496896538601961331976815719, −8.977261645849359773452334838514, −8.873291497111612933846721136141, −8.139892417713389916884358620360, −8.104087916944975258131967421443, −7.46307234656750136158883553530, −7.33673649050853434985665087803, −6.70570142232194579444992061677, −6.25636526933740798606536198205, −5.62414970792290860843230946545, −5.48036594909140966086025964907, −4.64393832626075969475875135623, −4.52825377038217125519558597020, −3.69832807006811576990550525925, −3.47647764324876054293851957952, −2.67381450202748859838066339126, −2.48332182477886635370329340956, −1.06727437927036061854231765705, −0.856868779798263474989688419630,
0.856868779798263474989688419630, 1.06727437927036061854231765705, 2.48332182477886635370329340956, 2.67381450202748859838066339126, 3.47647764324876054293851957952, 3.69832807006811576990550525925, 4.52825377038217125519558597020, 4.64393832626075969475875135623, 5.48036594909140966086025964907, 5.62414970792290860843230946545, 6.25636526933740798606536198205, 6.70570142232194579444992061677, 7.33673649050853434985665087803, 7.46307234656750136158883553530, 8.104087916944975258131967421443, 8.139892417713389916884358620360, 8.873291497111612933846721136141, 8.977261645849359773452334838514, 9.658496896538601961331976815719, 10.18467944199865474120508380945
