| L(s) = 1 | − 6·5-s + 16·7-s − 34·11-s + 24·13-s + 8·17-s − 46·23-s + 11·25-s + 50·31-s − 96·35-s + 18·37-s + 146·41-s − 78·43-s − 90·47-s + 128·49-s + 16·53-s + 204·55-s − 192·61-s − 144·65-s − 134·67-s − 50·71-s + 80·73-s − 544·77-s + 150·83-s − 48·85-s + 384·91-s − 186·97-s − 386·101-s + ⋯ |
| L(s) = 1 | − 6/5·5-s + 16/7·7-s − 3.09·11-s + 1.84·13-s + 8/17·17-s − 2·23-s + 0.439·25-s + 1.61·31-s − 2.74·35-s + 0.486·37-s + 3.56·41-s − 1.81·43-s − 1.91·47-s + 2.61·49-s + 0.301·53-s + 3.70·55-s − 3.14·61-s − 2.21·65-s − 2·67-s − 0.704·71-s + 1.09·73-s − 7.06·77-s + 1.80·83-s − 0.564·85-s + 4.21·91-s − 1.91·97-s − 3.82·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.010935708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010935708\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 239 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 1681 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 90 T + 4050 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4561 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 80 T + 3200 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8126 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T + 11250 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12817 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 186 T + 17298 T^{2} + 186 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−9.738129180411592991650752842295, −8.620886650615690620670490033101, −8.333127639248598075269789346994, −8.062739035922739414526819750438, −7.992120336733660171233099427134, −7.61359609506739031348141463400, −7.50050450229892619324338085094, −6.49794999237720636566711842020, −6.01470296298214817324981199098, −5.73600369926117380760189168273, −5.23751287318780789638140234486, −4.68530997351866776914959610296, −4.56879163949806733654733765267, −4.05455310825655844285712077072, −3.51298048439125993830523030487, −2.69326526304964460586577560142, −2.60435505726231424178531249327, −1.60956713113893411119346962396, −1.29483368525883241912780721627, −0.27989928432204751431434974277,
0.27989928432204751431434974277, 1.29483368525883241912780721627, 1.60956713113893411119346962396, 2.60435505726231424178531249327, 2.69326526304964460586577560142, 3.51298048439125993830523030487, 4.05455310825655844285712077072, 4.56879163949806733654733765267, 4.68530997351866776914959610296, 5.23751287318780789638140234486, 5.73600369926117380760189168273, 6.01470296298214817324981199098, 6.49794999237720636566711842020, 7.50050450229892619324338085094, 7.61359609506739031348141463400, 7.992120336733660171233099427134, 8.062739035922739414526819750438, 8.333127639248598075269789346994, 8.620886650615690620670490033101, 9.738129180411592991650752842295
