| L(s) = 1 | + 4·3-s + 4·7-s + 8·9-s + 16·11-s − 6·13-s + 14·17-s + 16·21-s − 4·23-s − 25·25-s + 36·27-s + 104·31-s + 64·33-s + 6·37-s − 24·39-s − 16·41-s + 84·43-s − 36·47-s + 8·49-s + 56·51-s − 106·53-s + 96·61-s + 32·63-s − 124·67-s − 16·69-s − 56·71-s − 94·73-s − 100·75-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 4/7·7-s + 8/9·9-s + 1.45·11-s − 0.461·13-s + 0.823·17-s + 0.761·21-s − 0.173·23-s − 25-s + 4/3·27-s + 3.35·31-s + 1.93·33-s + 6/37·37-s − 0.615·39-s − 0.390·41-s + 1.95·43-s − 0.765·47-s + 8/49·49-s + 1.09·51-s − 2·53-s + 1.57·61-s + 0.507·63-s − 1.85·67-s − 0.231·69-s − 0.788·71-s − 1.28·73-s − 4/3·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.521283195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.521283195\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 124 T + 7688 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.64607882151072088403009525243, −11.39327676336287180249698022478, −10.51892705650010151059998770332, −10.14479492757461918283613385651, −9.680039127101922974825941665738, −9.328045504588263865534143261686, −8.739644392277153183017969501821, −8.398460943915667755131748714658, −7.75152450861019584063470430515, −7.74916034382571295725625967703, −6.76985651513538834443181265468, −6.44606412800501350919894234585, −5.81786934139138164078574115614, −4.98333481203219928433741495413, −4.24787538765960950245126687354, −4.12905377665079554164938187557, −2.97789672215722418063483479817, −2.83173947653165560421207226790, −1.73673106186176142693196532384, −1.05555449912427093422493478929,
1.05555449912427093422493478929, 1.73673106186176142693196532384, 2.83173947653165560421207226790, 2.97789672215722418063483479817, 4.12905377665079554164938187557, 4.24787538765960950245126687354, 4.98333481203219928433741495413, 5.81786934139138164078574115614, 6.44606412800501350919894234585, 6.76985651513538834443181265468, 7.74916034382571295725625967703, 7.75152450861019584063470430515, 8.398460943915667755131748714658, 8.739644392277153183017969501821, 9.328045504588263865534143261686, 9.680039127101922974825941665738, 10.14479492757461918283613385651, 10.51892705650010151059998770332, 11.39327676336287180249698022478, 11.64607882151072088403009525243
