L(s) = 1 | + 8·7-s − 2·9-s − 2·17-s − 10·25-s + 8·31-s + 12·41-s + 34·49-s − 16·63-s + 4·73-s − 16·79-s − 5·81-s − 12·89-s + 28·97-s + 32·103-s − 12·113-s − 16·119-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 3.02·7-s − 2/3·9-s − 0.485·17-s − 2·25-s + 1.43·31-s + 1.87·41-s + 34/7·49-s − 2.01·63-s + 0.468·73-s − 1.80·79-s − 5/9·81-s − 1.27·89-s + 2.84·97-s + 3.15·103-s − 1.12·113-s − 1.46·119-s + 1.27·121-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 + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.457301172\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.457301172\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.763699502340737285319959905301, −8.241072165068691921287501783548, −8.020764485917199688918114818768, −7.56004329429510605274038490355, −7.25002689502959545343122319948, −6.35150301548018653136035247695, −5.66883563130032604301217732020, −5.66320470137327715389850473544, −4.65035915337384466130876980903, −4.61798148851481694877863510515, −4.09840146621603381404148858714, −3.14432806107716120656228384149, −2.18485014981353314837233550045, −1.95520344137895444953340586126, −0.984825652924800667366751863259,
0.984825652924800667366751863259, 1.95520344137895444953340586126, 2.18485014981353314837233550045, 3.14432806107716120656228384149, 4.09840146621603381404148858714, 4.61798148851481694877863510515, 4.65035915337384466130876980903, 5.66320470137327715389850473544, 5.66883563130032604301217732020, 6.35150301548018653136035247695, 7.25002689502959545343122319948, 7.56004329429510605274038490355, 8.020764485917199688918114818768, 8.241072165068691921287501783548, 8.763699502340737285319959905301