L(s) = 1 | − 3-s + 0.511·7-s + 9-s − 1.82·11-s − 6.12·13-s − 2.58·17-s − 4.86·19-s − 0.511·21-s + 6.63·23-s − 27-s − 7.99·29-s − 4.88·31-s + 1.82·33-s + 7.43·37-s + 6.12·39-s + 5.73·41-s + 2.05·43-s − 7.95·47-s − 6.73·49-s + 2.58·51-s − 1.32·53-s + 4.86·57-s − 10.0·59-s + 4.77·61-s + 0.511·63-s + 14.3·67-s − 6.63·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.193·7-s + 0.333·9-s − 0.550·11-s − 1.69·13-s − 0.626·17-s − 1.11·19-s − 0.111·21-s + 1.38·23-s − 0.192·27-s − 1.48·29-s − 0.877·31-s + 0.318·33-s + 1.22·37-s + 0.980·39-s + 0.894·41-s + 0.312·43-s − 1.16·47-s − 0.962·49-s + 0.361·51-s − 0.182·53-s + 0.644·57-s − 1.31·59-s + 0.610·61-s + 0.0644·63-s + 1.75·67-s − 0.798·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8403342986\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8403342986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.511T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 4.86T + 19T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 + 7.95T + 47T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 4.77T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 + 0.912T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
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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
−7.61954863491987478626261078212, −7.32081343977713016555585358533, −6.48418760529711862473624373647, −5.78822640957119362109997544795, −4.83589406701961187282582026775, −4.72818947000473391014263849979, −3.58480545762820453162152105392, −2.53332260651296412867119156117, −1.89659819478581458218814480479, −0.45119408251613334591219694519,
0.45119408251613334591219694519, 1.89659819478581458218814480479, 2.53332260651296412867119156117, 3.58480545762820453162152105392, 4.72818947000473391014263849979, 4.83589406701961187282582026775, 5.78822640957119362109997544795, 6.48418760529711862473624373647, 7.32081343977713016555585358533, 7.61954863491987478626261078212