L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s + 11-s + 4·13-s − 2·15-s + 4·19-s − 8·21-s − 6·23-s + 25-s − 4·27-s + 6·29-s + 8·31-s + 2·33-s + 4·35-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 45-s + 6·47-s + 9·49-s + 6·53-s − 55-s + 8·57-s + 12·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.516·15-s + 0.917·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.348·33-s + 0.676·35-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s + 1.05·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.206561967\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.206561967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.665100632600319263086471254534, −7.972421750206537781990244124247, −7.18400363171512990535747166596, −6.36019171365228631498930493799, −5.81412801894900173737519471646, −4.43187562216518006123836531561, −3.57217047624157982849034967897, −3.22125682695351147788730933130, −2.29840629560872239152877430188, −0.821901278182008429108946759389,
0.821901278182008429108946759389, 2.29840629560872239152877430188, 3.22125682695351147788730933130, 3.57217047624157982849034967897, 4.43187562216518006123836531561, 5.81412801894900173737519471646, 6.36019171365228631498930493799, 7.18400363171512990535747166596, 7.972421750206537781990244124247, 8.665100632600319263086471254534