| L(s) = 1 | + 7·16-s − 32·31-s + 8·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 7/4·16-s − 5.74·31-s + 1.02·61-s − 81-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7442011239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7442011239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 382 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 98 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 1778 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 9938 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76081486330043137405137605251, −10.53061872040027379540653170350, −10.41457947495543246540209379939, −9.819690080857000099844974709186, −9.709095682617365801481477083698, −9.229114639916129374277969711062, −9.186000810889273437221384996387, −8.635593233161787413572018965313, −8.509438765201328517434046221591, −8.030321890646285451268884181389, −7.69629943029747480888817969025, −7.34620001755066011427968295329, −7.07167264269865822101784048619, −7.03391025821551306767908559308, −6.14944178014170683784485800338, −5.97508446664511383909886609544, −5.59619503202360186135513635054, −5.23175479693293312759926761244, −5.11986073900510071403803114577, −4.21091389250569490413296196158, −3.93615513183635762614798187508, −3.34831099937780030944362603971, −3.28819910129529287773214584801, −2.17607389815220050882504286441, −1.64009472245150218811766256515,
1.64009472245150218811766256515, 2.17607389815220050882504286441, 3.28819910129529287773214584801, 3.34831099937780030944362603971, 3.93615513183635762614798187508, 4.21091389250569490413296196158, 5.11986073900510071403803114577, 5.23175479693293312759926761244, 5.59619503202360186135513635054, 5.97508446664511383909886609544, 6.14944178014170683784485800338, 7.03391025821551306767908559308, 7.07167264269865822101784048619, 7.34620001755066011427968295329, 7.69629943029747480888817969025, 8.030321890646285451268884181389, 8.509438765201328517434046221591, 8.635593233161787413572018965313, 9.186000810889273437221384996387, 9.229114639916129374277969711062, 9.709095682617365801481477083698, 9.819690080857000099844974709186, 10.41457947495543246540209379939, 10.53061872040027379540653170350, 10.76081486330043137405137605251
