| L(s) = 1 | + 3-s − 5-s − 7-s − 11-s + 13-s − 15-s + 17-s − 21-s + 25-s − 27-s − 29-s − 33-s + 35-s + 39-s + 47-s + 49-s + 51-s + 55-s − 65-s + 2·71-s − 2·73-s + 75-s + 77-s − 79-s − 81-s − 2·83-s − 85-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s − 7-s − 11-s + 13-s − 15-s + 17-s − 21-s + 25-s − 27-s − 29-s − 33-s + 35-s + 39-s + 47-s + 49-s + 51-s + 55-s − 65-s + 2·71-s − 2·73-s + 75-s + 77-s − 79-s − 81-s − 2·83-s − 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6671566555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6671566555\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−13.36828070088032391983780080948, −12.61389508552208695184395621109, −11.39324135528712356979884961047, −10.26619992550177601441111145444, −9.070013161477823014534903801094, −8.171028891246932360218079562191, −7.30305633226858282819250127281, −5.72397637894988209050924902995, −3.82211446752265937719735687028, −2.94378740484957857899261155773,
2.94378740484957857899261155773, 3.82211446752265937719735687028, 5.72397637894988209050924902995, 7.30305633226858282819250127281, 8.171028891246932360218079562191, 9.070013161477823014534903801094, 10.26619992550177601441111145444, 11.39324135528712356979884961047, 12.61389508552208695184395621109, 13.36828070088032391983780080948
