| L(s) = 1 | + 7-s + 2·9-s + 11-s − 13-s + 19-s + 23-s − 37-s − 41-s + 47-s − 53-s + 59-s + 2·63-s + 77-s + 3·81-s − 89-s − 91-s + 2·99-s + 103-s − 2·117-s + 127-s + 131-s + 133-s + 137-s + 139-s − 143-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 7-s + 2·9-s + 11-s − 13-s + 19-s + 23-s − 37-s − 41-s + 47-s − 53-s + 59-s + 2·63-s + 77-s + 3·81-s − 89-s − 91-s + 2·99-s + 103-s − 2·117-s + 127-s + 131-s + 133-s + 137-s + 139-s − 143-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.375393890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375393890\) |
| \(L(1)\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 53 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 59 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.881577073493336601636099886802, −8.414515600990170755986917714056, −7.931459427655947835570423182687, −7.70719294133735193083891291819, −7.31802313771511103935102825699, −6.99766707196473639624848527592, −6.69611018819678621099816586956, −6.53507336263192474374206149002, −5.72535084843712405811691345050, −5.27306332124088792707038288570, −5.10791005605023283270777100268, −4.65348080164546101045168717668, −4.19208158143370062057133238775, −4.10948623707712087799111624458, −3.26463002490018765252086246414, −3.19483955427911074583385104823, −2.20705075468631241711432618609, −1.92983358433021845031650981061, −1.30719927401168804551636563848, −1.04328909854902685539345275255,
1.04328909854902685539345275255, 1.30719927401168804551636563848, 1.92983358433021845031650981061, 2.20705075468631241711432618609, 3.19483955427911074583385104823, 3.26463002490018765252086246414, 4.10948623707712087799111624458, 4.19208158143370062057133238775, 4.65348080164546101045168717668, 5.10791005605023283270777100268, 5.27306332124088792707038288570, 5.72535084843712405811691345050, 6.53507336263192474374206149002, 6.69611018819678621099816586956, 6.99766707196473639624848527592, 7.31802313771511103935102825699, 7.70719294133735193083891291819, 7.931459427655947835570423182687, 8.414515600990170755986917714056, 8.881577073493336601636099886802
