| L(s) = 1 | + 3-s − 2·7-s − 9-s + 2·11-s + 2·13-s − 4·17-s + 8·19-s − 2·21-s + 9·23-s − 2·29-s + 7·31-s + 2·33-s + 11·37-s + 2·39-s + 6·41-s − 6·43-s + 16·47-s + 6·49-s − 4·51-s − 8·53-s + 8·57-s + 5·59-s − 6·61-s + 2·63-s + 15·67-s + 9·69-s + 5·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s − 0.436·21-s + 1.87·23-s − 0.371·29-s + 1.25·31-s + 0.348·33-s + 1.80·37-s + 0.320·39-s + 0.937·41-s − 0.914·43-s + 2.33·47-s + 6/7·49-s − 0.560·51-s − 1.09·53-s + 1.05·57-s + 0.650·59-s − 0.768·61-s + 0.251·63-s + 1.83·67-s + 1.08·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.487637720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.487637720\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15 T + 186 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 372 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
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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
−8.569294575681776858219842774550, −8.281387559201828740614377237443, −7.75146244408403559912236551496, −7.50716993787751304033219123967, −7.10375252011538161002163025072, −6.69478767901061623965317475304, −6.39089513160018131270292765758, −6.15348180155387446570021420381, −5.44481374385695821500459405959, −5.37944320317922370394141190915, −4.80677791670964600603505839472, −4.30205099804158779536748918872, −3.95161211206438174441005613083, −3.56694046126998976573294062261, −2.94508075298548520732699765524, −2.82462795893967783044342484210, −2.45923839405037145469780008008, −1.66051572810067403879634525651, −0.836455123158833036295590329061, −0.821154470808708518466482035081,
0.821154470808708518466482035081, 0.836455123158833036295590329061, 1.66051572810067403879634525651, 2.45923839405037145469780008008, 2.82462795893967783044342484210, 2.94508075298548520732699765524, 3.56694046126998976573294062261, 3.95161211206438174441005613083, 4.30205099804158779536748918872, 4.80677791670964600603505839472, 5.37944320317922370394141190915, 5.44481374385695821500459405959, 6.15348180155387446570021420381, 6.39089513160018131270292765758, 6.69478767901061623965317475304, 7.10375252011538161002163025072, 7.50716993787751304033219123967, 7.75146244408403559912236551496, 8.281387559201828740614377237443, 8.569294575681776858219842774550
