| L(s) = 1 | − 3-s − 3.93i·5-s − 1.78i·7-s + 9-s − 4.72i·11-s + 3.93i·15-s − 0.784·17-s + 5.15i·19-s + 1.78i·21-s − 4.72·23-s − 10.5·25-s − 27-s − 7.93·29-s − 2.93i·31-s + 4.72i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.76i·5-s − 0.674i·7-s + 0.333·9-s − 1.42i·11-s + 1.01i·15-s − 0.190·17-s + 1.18i·19-s + 0.389i·21-s − 0.984·23-s − 2.10·25-s − 0.192·27-s − 1.47·29-s − 0.527i·31-s + 0.822i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5605490679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5605490679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.93iT - 5T^{2} \) |
| 7 | \( 1 + 1.78iT - 7T^{2} \) |
| 11 | \( 1 + 4.72iT - 11T^{2} \) |
| 17 | \( 1 + 0.784T + 17T^{2} \) |
| 19 | \( 1 - 5.15iT - 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 2.93iT - 31T^{2} \) |
| 37 | \( 1 + 1.21iT - 37T^{2} \) |
| 41 | \( 1 + 8.66iT - 41T^{2} \) |
| 43 | \( 1 - 6.21T + 43T^{2} \) |
| 47 | \( 1 + 0.722iT - 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 0.430iT - 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 - 9.78iT - 67T^{2} \) |
| 71 | \( 1 - 8.72iT - 71T^{2} \) |
| 73 | \( 1 + 4.87iT - 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.59iT - 83T^{2} \) |
| 89 | \( 1 + 15.8iT - 89T^{2} \) |
| 97 | \( 1 - 3.06iT - 97T^{2} \) |
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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
−7.988018390933065009402080067086, −7.43338108997601276239733145584, −6.12027604106664449059643742136, −5.74824698288310188689855299510, −5.06718551387162190875249865797, −4.03796807114561316068123306031, −3.74036745745582756835684696964, −1.97388833078436812896393461025, −1.00815118118037333445801951871, −0.19414602813226873201606674796,
1.89505646226640339170563077950, 2.51655977148635525774327986799, 3.46193753214097762221843232993, 4.43800403356947925677729824843, 5.24626332822674003910963630763, 6.25175712306596357579574087631, 6.57758730395580813388391003337, 7.40868054009777173157727856559, 7.81820760423996115650836389321, 9.171036959149006812983112402771
