| L(s) = 1 | − 1.41·3-s − 14.1·5-s − 25·9-s − 54·11-s − 22.6·13-s + 20.0·15-s − 32.5·17-s − 63.6·19-s − 28·23-s + 75.0·25-s + 73.5·27-s + 282·29-s − 274.·31-s + 76.3·33-s + 146·37-s + 32.0·39-s − 340.·41-s − 10·43-s + 353.·45-s − 506.·47-s + 46·51-s + 598·53-s + 763.·55-s + 90.0·57-s + 575.·59-s − 466.·61-s + 320.·65-s + ⋯ |
| L(s) = 1 | − 0.272·3-s − 1.26·5-s − 0.925·9-s − 1.48·11-s − 0.482·13-s + 0.344·15-s − 0.464·17-s − 0.768·19-s − 0.253·23-s + 0.600·25-s + 0.524·27-s + 1.80·29-s − 1.58·31-s + 0.402·33-s + 0.648·37-s + 0.131·39-s − 1.29·41-s − 0.0354·43-s + 1.17·45-s − 1.57·47-s + 0.126·51-s + 1.54·53-s + 1.87·55-s + 0.209·57-s + 1.27·59-s − 0.979·61-s + 0.610·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3265589140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3265589140\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 27T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 11 | \( 1 + 54T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282T + 2.43e4T^{2} \) |
| 31 | \( 1 + 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 146T + 5.06e4T^{2} \) |
| 41 | \( 1 + 340.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 10T + 7.95e4T^{2} \) |
| 47 | \( 1 + 506.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 598T + 1.48e5T^{2} \) |
| 59 | \( 1 - 575.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 916T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420T + 3.57e5T^{2} \) |
| 73 | \( 1 - 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 292T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 445.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 589.T + 9.12e5T^{2} \) |
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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
−10.14425147321444999456320891545, −8.765295738027926042768324824707, −8.189360181299438027138228347656, −7.46439319181728701874313839769, −6.44440957923723176584907160561, −5.30960878219809709969523597726, −4.55615484986022938109523938075, −3.36051883843754759910814119438, −2.37244683792370218740736535234, −0.30254483311104846866936487384,
0.30254483311104846866936487384, 2.37244683792370218740736535234, 3.36051883843754759910814119438, 4.55615484986022938109523938075, 5.30960878219809709969523597726, 6.44440957923723176584907160561, 7.46439319181728701874313839769, 8.189360181299438027138228347656, 8.765295738027926042768324824707, 10.14425147321444999456320891545
