| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s − 15·9-s − 4·12-s − 36·13-s − 4·16-s − 24·17-s − 30·18-s − 12·19-s − 36·23-s − 25·25-s − 72·26-s + 50·27-s + 48·29-s + 2·31-s − 8·32-s − 48·34-s − 30·36-s − 24·37-s − 24·38-s + 72·39-s + 14·43-s − 72·46-s + 8·48-s + 73·49-s + ⋯ |
| L(s) = 1 | + 2-s − 2/3·3-s + 1/2·4-s − 2/3·6-s − 5/3·9-s − 1/3·12-s − 2.76·13-s − 1/4·16-s − 1.41·17-s − 5/3·18-s − 0.631·19-s − 1.56·23-s − 25-s − 2.76·26-s + 1.85·27-s + 1.65·29-s + 2/31·31-s − 1/4·32-s − 1.41·34-s − 5/6·36-s − 0.648·37-s − 0.631·38-s + 1.84·39-s + 0.325·43-s − 1.56·46-s + 1/6·48-s + 1.48·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.006555922704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006555922704\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 37 | $C_2$ | \( 1 + 24 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 217 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2737 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 132 T + 8712 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 84 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1847 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 118 T + 6962 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 74 T + 2738 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.00572434956376667688686009063, −10.94575211606181011592961659810, −10.66789253589742450202112302917, −10.21754388574543515078671063241, −9.675130502415073239903184587758, −8.935019432513740777514929460444, −8.853966726919996419356081500405, −7.84090216801500269545446783617, −7.81355734386059063153092741568, −6.85818562647205865931814377754, −6.55100290395304007361033098191, −5.82167398377426007417716241558, −5.74511816218498224547558824199, −4.80349143253818085474934985385, −4.71548274537197840012526881346, −4.10481313048168081777939449241, −3.05897630958603433689493058524, −2.53357473985066487157227108296, −2.13501186129939169164701278747, −0.02910231835641983130373418624,
0.02910231835641983130373418624, 2.13501186129939169164701278747, 2.53357473985066487157227108296, 3.05897630958603433689493058524, 4.10481313048168081777939449241, 4.71548274537197840012526881346, 4.80349143253818085474934985385, 5.74511816218498224547558824199, 5.82167398377426007417716241558, 6.55100290395304007361033098191, 6.85818562647205865931814377754, 7.81355734386059063153092741568, 7.84090216801500269545446783617, 8.853966726919996419356081500405, 8.935019432513740777514929460444, 9.675130502415073239903184587758, 10.21754388574543515078671063241, 10.66789253589742450202112302917, 10.94575211606181011592961659810, 12.00572434956376667688686009063
