| L(s) = 1 | + 2·5-s + 9-s + 4·11-s − 8·23-s − 25-s + 8·31-s + 8·37-s + 2·45-s − 16·47-s + 6·49-s + 8·55-s − 8·59-s + 8·67-s + 81-s + 20·89-s + 24·97-s + 4·99-s + 8·103-s + 16·113-s − 16·115-s + 5·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.66·23-s − 1/5·25-s + 1.43·31-s + 1.31·37-s + 0.298·45-s − 2.33·47-s + 6/7·49-s + 1.07·55-s − 1.04·59-s + 0.977·67-s + 1/9·81-s + 2.11·89-s + 2.43·97-s + 0.402·99-s + 0.788·103-s + 1.50·113-s − 1.49·115-s + 5/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.928391727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.928391727\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69571962038807276031370315754, −7.59881995572894193573191634931, −6.81031582604890326154745677544, −6.34930711160599876332787871817, −6.16690703951277200294196417096, −5.97753391701587032971561910054, −5.10768274118646309202684633907, −4.80256537847599016115727134079, −4.25899809688276604609139833133, −3.80287348591858534222642664302, −3.32749766678586426022421231129, −2.57069968709100569063497016289, −2.01165689549822344639516080132, −1.56014447875272277226544957043, −0.73516088596436919909002249258,
0.73516088596436919909002249258, 1.56014447875272277226544957043, 2.01165689549822344639516080132, 2.57069968709100569063497016289, 3.32749766678586426022421231129, 3.80287348591858534222642664302, 4.25899809688276604609139833133, 4.80256537847599016115727134079, 5.10768274118646309202684633907, 5.97753391701587032971561910054, 6.16690703951277200294196417096, 6.34930711160599876332787871817, 6.81031582604890326154745677544, 7.59881995572894193573191634931, 7.69571962038807276031370315754
