| L(s) = 1 | − 3-s − 5-s + 9-s + 4.82·11-s + 15-s + 2.82·17-s − 1.41·19-s + 0.585·23-s + 25-s − 27-s − 2·29-s − 2.58·31-s − 4.82·33-s + 8.82·37-s − 11.6·41-s − 5.65·43-s − 45-s + 4.82·47-s − 2.82·51-s + 3.41·53-s − 4.82·55-s + 1.41·57-s + 8.48·59-s − 7.07·61-s + 1.17·67-s − 0.585·69-s − 3.65·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.333·9-s + 1.45·11-s + 0.258·15-s + 0.685·17-s − 0.324·19-s + 0.122·23-s + 0.200·25-s − 0.192·27-s − 0.371·29-s − 0.464·31-s − 0.840·33-s + 1.45·37-s − 1.82·41-s − 0.862·43-s − 0.149·45-s + 0.704·47-s − 0.396·51-s + 0.468·53-s − 0.651·55-s + 0.187·57-s + 1.10·59-s − 0.905·61-s + 0.143·67-s − 0.0705·69-s − 0.433·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.452640151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.452640151\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 8.82T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 3.41T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 0.828T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−8.768123528425516953465188937320, −7.975479670057227601828315367190, −7.11226564184344149187521705841, −6.52492231158284317036693811095, −5.74818191541906650226273083985, −4.85407905997111063045792523402, −4.00758163360049015526141641266, −3.33837955470237877306197354373, −1.88001459618901922487058289703, −0.78608392548294445264747606666,
0.78608392548294445264747606666, 1.88001459618901922487058289703, 3.33837955470237877306197354373, 4.00758163360049015526141641266, 4.85407905997111063045792523402, 5.74818191541906650226273083985, 6.52492231158284317036693811095, 7.11226564184344149187521705841, 7.975479670057227601828315367190, 8.768123528425516953465188937320
