| L(s) = 1 | + 0.445·3-s + 1.24·5-s − 0.801·9-s + 0.554·15-s + 1.80·19-s + 0.554·25-s − 0.801·27-s − 0.445·29-s − 1.80·37-s − 1.24·43-s − 45-s + 49-s + 0.801·57-s − 71-s + 1.24·73-s + 0.246·75-s − 1.24·79-s + 0.445·81-s + 1.80·83-s − 0.198·87-s − 0.445·89-s + 2.24·95-s − 1.80·101-s + 0.445·103-s − 2·107-s − 0.445·109-s − 0.801·111-s + ⋯ |
| L(s) = 1 | + 0.445·3-s + 1.24·5-s − 0.801·9-s + 0.554·15-s + 1.80·19-s + 0.554·25-s − 0.801·27-s − 0.445·29-s − 1.80·37-s − 1.24·43-s − 45-s + 49-s + 0.801·57-s − 71-s + 1.24·73-s + 0.246·75-s − 1.24·79-s + 0.445·81-s + 1.80·83-s − 0.198·87-s − 0.445·89-s + 2.24·95-s − 1.80·101-s + 0.445·103-s − 2·107-s − 0.445·109-s − 0.801·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.399578555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.399578555\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 3 | \( 1 - 0.445T + T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 + 1.24T + T^{2} \) |
| 83 | \( 1 - 1.80T + T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 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
−9.835899700566668051745674772059, −9.269005759012636100121515792019, −8.538730368206714819197520039026, −7.57661524302358846100437895949, −6.64218126106786124604289641631, −5.61074597004115415407295561827, −5.20310848996734236438145051772, −3.60596893121611429332383702272, −2.71658165374624894194217573694, −1.62866880998304416213212717173,
1.62866880998304416213212717173, 2.71658165374624894194217573694, 3.60596893121611429332383702272, 5.20310848996734236438145051772, 5.61074597004115415407295561827, 6.64218126106786124604289641631, 7.57661524302358846100437895949, 8.538730368206714819197520039026, 9.269005759012636100121515792019, 9.835899700566668051745674772059
