| L(s) = 1 | − 30·11-s + 18·23-s + 47·25-s + 12·29-s + 62·37-s + 20·43-s + 114·53-s − 98·67-s + 252·71-s − 146·79-s − 78·107-s + 206·109-s + 156·113-s + 433·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 146·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.72·11-s + 0.782·23-s + 1.87·25-s + 0.413·29-s + 1.67·37-s + 0.465·43-s + 2.15·53-s − 1.46·67-s + 3.54·71-s − 1.84·79-s − 0.728·107-s + 1.88·109-s + 1.38·113-s + 3.57·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.863·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.664702862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.664702862\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 47 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 15 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 479 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1775 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 31 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2543 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 57 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 815 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9983 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12575 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18050 T^{2} + p^{4} T^{4} \) |
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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
−9.266516592920165519109020750299, −8.825597771290187786059022831184, −8.460099871420494628764283784454, −8.199350390051601648190330111268, −7.68719600200585764960468314323, −7.39659437130214176517614305994, −7.04491463729357615551467459969, −6.57767366075665151556226181051, −6.00344629399554636227961231826, −5.61687491560076589869036013951, −5.12820042806096159824278468304, −4.98556198842506809151899971711, −4.43964661081243218293905152280, −3.95985196636720081955199478698, −3.04109716624866194436464504691, −2.92678658695066597240886218775, −2.49786020219107412704605661220, −1.92462222696428411671343487138, −0.832660343313386208158604761224, −0.57734704422095786527599113103,
0.57734704422095786527599113103, 0.832660343313386208158604761224, 1.92462222696428411671343487138, 2.49786020219107412704605661220, 2.92678658695066597240886218775, 3.04109716624866194436464504691, 3.95985196636720081955199478698, 4.43964661081243218293905152280, 4.98556198842506809151899971711, 5.12820042806096159824278468304, 5.61687491560076589869036013951, 6.00344629399554636227961231826, 6.57767366075665151556226181051, 7.04491463729357615551467459969, 7.39659437130214176517614305994, 7.68719600200585764960468314323, 8.199350390051601648190330111268, 8.460099871420494628764283784454, 8.825597771290187786059022831184, 9.266516592920165519109020750299
