| L(s) = 1 | + 5-s + 4·7-s − 6·11-s + 4·13-s − 6·17-s − 14·19-s + 9·23-s + 7·31-s + 4·35-s + 4·37-s − 6·41-s − 2·43-s + 7·49-s + 18·53-s − 6·55-s + 12·59-s + 7·61-s + 4·65-s − 2·67-s + 12·71-s + 4·73-s − 24·77-s + 79-s − 9·83-s − 6·85-s − 12·89-s + 16·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.80·11-s + 1.10·13-s − 1.45·17-s − 3.21·19-s + 1.87·23-s + 1.25·31-s + 0.676·35-s + 0.657·37-s − 0.937·41-s − 0.304·43-s + 49-s + 2.47·53-s − 0.809·55-s + 1.56·59-s + 0.896·61-s + 0.496·65-s − 0.244·67-s + 1.42·71-s + 0.468·73-s − 2.73·77-s + 0.112·79-s − 0.987·83-s − 0.650·85-s − 1.27·89-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.257898644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257898644\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.687200913111515530652736619547, −8.757493628001891471760565181431, −8.752534197138459485129551515238, −8.512739962264750531376093100781, −8.109249006716829999497575186053, −7.87261579882732902289032224544, −6.98470965588610863681247215939, −6.77610104135923768563833277775, −6.54856552294696196246227120948, −5.85750444921287882927878135131, −5.30000381175256296814463956878, −5.24293915020235749807143053781, −4.49577476684186458958711968611, −4.32701450021317210206958952492, −3.87338580198705610077668300770, −2.89164589802610923783387718342, −2.35606617907606093409337147392, −2.23904887785376749163383947650, −1.48194867072812770596460442739, −0.57632350635446817820360542626,
0.57632350635446817820360542626, 1.48194867072812770596460442739, 2.23904887785376749163383947650, 2.35606617907606093409337147392, 2.89164589802610923783387718342, 3.87338580198705610077668300770, 4.32701450021317210206958952492, 4.49577476684186458958711968611, 5.24293915020235749807143053781, 5.30000381175256296814463956878, 5.85750444921287882927878135131, 6.54856552294696196246227120948, 6.77610104135923768563833277775, 6.98470965588610863681247215939, 7.87261579882732902289032224544, 8.109249006716829999497575186053, 8.512739962264750531376093100781, 8.752534197138459485129551515238, 8.757493628001891471760565181431, 9.687200913111515530652736619547
