L(s) = 1 | − 1.14·3-s + 4.20·5-s − 0.775·7-s − 1.68·9-s + 11-s + 4.03·13-s − 4.82·15-s − 4.22·17-s + 2.69·19-s + 0.889·21-s + 6.08·23-s + 12.6·25-s + 5.37·27-s − 3.27·29-s + 3.97·31-s − 1.14·33-s − 3.26·35-s − 1.35·37-s − 4.63·39-s − 2.86·41-s − 4.82·43-s − 7.08·45-s + 3.30·47-s − 6.39·49-s + 4.85·51-s + 2.03·53-s + 4.20·55-s + ⋯ |
L(s) = 1 | − 0.662·3-s + 1.88·5-s − 0.293·7-s − 0.561·9-s + 0.301·11-s + 1.11·13-s − 1.24·15-s − 1.02·17-s + 0.618·19-s + 0.194·21-s + 1.26·23-s + 2.53·25-s + 1.03·27-s − 0.608·29-s + 0.713·31-s − 0.199·33-s − 0.551·35-s − 0.222·37-s − 0.741·39-s − 0.446·41-s − 0.736·43-s − 1.05·45-s + 0.481·47-s − 0.914·49-s + 0.679·51-s + 0.279·53-s + 0.567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.327736484\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.327736484\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
good | 3 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 - 4.20T + 5T^{2} \) |
| 7 | \( 1 + 0.775T + 7T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 + 4.22T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 6.08T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 1.35T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 + 8.49T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 0.849T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 8.06T + 83T^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 + 8.93T + 97T^{2} \) |
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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
−8.280292193311296402466975378819, −6.96772682760248020199768723744, −6.44047629724127132876992758657, −6.05773017488966328211797596608, −5.29284466766613755986749570378, −4.82742690008468686694000752324, −3.49234935690724006586656049721, −2.70084178535577668747212779404, −1.76858490176776999962342343115, −0.862860558843668834863853367868,
0.862860558843668834863853367868, 1.76858490176776999962342343115, 2.70084178535577668747212779404, 3.49234935690724006586656049721, 4.82742690008468686694000752324, 5.29284466766613755986749570378, 6.05773017488966328211797596608, 6.44047629724127132876992758657, 6.96772682760248020199768723744, 8.280292193311296402466975378819