L(s) = 1 | + 3-s − 7-s − 2·9-s + 6·11-s − 5·13-s − 3·17-s − 19-s − 21-s + 3·23-s − 5·27-s + 9·29-s + 4·31-s + 6·33-s − 2·37-s − 5·39-s + 8·43-s − 6·49-s − 3·51-s + 3·53-s − 57-s − 9·59-s − 10·61-s + 2·63-s + 5·67-s + 3·69-s + 6·71-s + 7·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.80·11-s − 1.38·13-s − 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.625·23-s − 0.962·27-s + 1.67·29-s + 0.718·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s + 1.21·43-s − 6/7·49-s − 0.420·51-s + 0.412·53-s − 0.132·57-s − 1.17·59-s − 1.28·61-s + 0.251·63-s + 0.610·67-s + 0.361·69-s + 0.712·71-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.151325934\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.151325934\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.965660853998351532368982028444, −7.08971577668189492485989188394, −6.55874110484964269431451883562, −5.98727468337922828585960000348, −4.85431664116277716307335752712, −4.35446566947163584234256974578, −3.39515826505100747996509869374, −2.75515250851414006834070112474, −1.94400683778762803472021635957, −0.70045234573102504240688012324,
0.70045234573102504240688012324, 1.94400683778762803472021635957, 2.75515250851414006834070112474, 3.39515826505100747996509869374, 4.35446566947163584234256974578, 4.85431664116277716307335752712, 5.98727468337922828585960000348, 6.55874110484964269431451883562, 7.08971577668189492485989188394, 7.965660853998351532368982028444