| L(s) = 1 | + 3-s + 4.36·5-s + 0.320·7-s + 9-s + 2.75·11-s + 2.28·13-s + 4.36·15-s + 1.14·17-s − 5.29·19-s + 0.320·21-s − 23-s + 14.0·25-s + 27-s + 29-s + 10.4·31-s + 2.75·33-s + 1.40·35-s − 1.14·37-s + 2.28·39-s − 12.4·41-s + 2.09·43-s + 4.36·45-s + 13.3·47-s − 6.89·49-s + 1.14·51-s + 0.339·53-s + 12.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.95·5-s + 0.121·7-s + 0.333·9-s + 0.831·11-s + 0.632·13-s + 1.12·15-s + 0.277·17-s − 1.21·19-s + 0.0700·21-s − 0.208·23-s + 2.80·25-s + 0.192·27-s + 0.185·29-s + 1.87·31-s + 0.480·33-s + 0.236·35-s − 0.187·37-s + 0.365·39-s − 1.95·41-s + 0.319·43-s + 0.650·45-s + 1.95·47-s − 0.985·49-s + 0.160·51-s + 0.0466·53-s + 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.578388848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.578388848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 4.36T + 5T^{2} \) |
| 7 | \( 1 - 0.320T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 0.339T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 + 9.22T + 61T^{2} \) |
| 67 | \( 1 + 9.13T + 67T^{2} \) |
| 71 | \( 1 - 5.62T + 71T^{2} \) |
| 73 | \( 1 - 0.0921T + 73T^{2} \) |
| 79 | \( 1 - 1.45T + 79T^{2} \) |
| 83 | \( 1 - 0.226T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 - 4.65T + 97T^{2} \) |
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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
−8.005671486083561103197928691066, −6.91829063827564481512815940892, −6.30772297945550322195983200353, −6.03560415100563749046628991996, −5.02244402436469982322690479242, −4.36659519920889417615723142706, −3.34644033763562343640515123353, −2.53527766511659744523475565871, −1.79603607115654925566994295545, −1.14655553050587746503988781045,
1.14655553050587746503988781045, 1.79603607115654925566994295545, 2.53527766511659744523475565871, 3.34644033763562343640515123353, 4.36659519920889417615723142706, 5.02244402436469982322690479242, 6.03560415100563749046628991996, 6.30772297945550322195983200353, 6.91829063827564481512815940892, 8.005671486083561103197928691066
