| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 25-s − 2·29-s − 32-s + 36-s − 2·37-s + 50-s + 2·53-s + 2·58-s + 64-s − 72-s + 2·74-s + 81-s − 100-s − 2·106-s − 2·109-s + 2·113-s − 2·116-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 25-s − 2·29-s − 32-s + 36-s − 2·37-s + 50-s + 2·53-s + 2·58-s + 64-s − 72-s + 2·74-s + 81-s − 100-s − 2·106-s − 2·109-s + 2·113-s − 2·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4878156744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4878156744\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.53191722332769884863651571813, −11.60687140681657786082542786270, −10.56991880068010727437427229026, −9.778317079932505908388739199955, −8.854075717860665745713986608863, −7.66109559082232720382507376407, −6.90375754611187260449823775175, −5.56686603511860954656868685423, −3.75433284828030179910406176303, −1.88434292211208747400285521709,
1.88434292211208747400285521709, 3.75433284828030179910406176303, 5.56686603511860954656868685423, 6.90375754611187260449823775175, 7.66109559082232720382507376407, 8.854075717860665745713986608863, 9.778317079932505908388739199955, 10.56991880068010727437427229026, 11.60687140681657786082542786270, 12.53191722332769884863651571813
