| L(s) = 1 | + 3-s − 5-s + 9-s + 6·13-s − 15-s + 2·17-s + 23-s + 25-s + 27-s + 6·29-s − 8·31-s + 10·37-s + 6·39-s − 6·41-s + 8·43-s − 45-s − 8·47-s − 7·49-s + 2·51-s − 6·53-s + 4·59-s − 6·61-s − 6·65-s − 8·67-s + 69-s + 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.960·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 1.16·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.520·59-s − 0.768·61-s − 0.744·65-s − 0.977·67-s + 0.120·69-s + 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.572657568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.572657568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.019359359297544311096761708925, −7.75393713467786547327847976436, −6.66317179399237957120169035073, −6.17639171616166515298521856012, −5.19973102520078041679919841676, −4.33942834181879546583004669500, −3.56996289387197501612160070732, −3.04148862989307749653136785216, −1.83359373015065893554420547885, −0.874778452891290069414683902721,
0.874778452891290069414683902721, 1.83359373015065893554420547885, 3.04148862989307749653136785216, 3.56996289387197501612160070732, 4.33942834181879546583004669500, 5.19973102520078041679919841676, 6.17639171616166515298521856012, 6.66317179399237957120169035073, 7.75393713467786547327847976436, 8.019359359297544311096761708925
