| L(s) = 1 | + i·5-s − 3.88i·7-s + 1.54i·11-s + (3.54 − 0.660i)13-s − 2.86i·19-s − 5.42·23-s − 25-s + 5.20·29-s − 6.22i·31-s + 3.88·35-s + 8.56i·37-s − 9.08i·41-s + 0.980·43-s − 6.52i·47-s − 8.08·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 1.46i·7-s + 0.465i·11-s + (0.983 − 0.183i)13-s − 0.657i·19-s − 1.13·23-s − 0.200·25-s + 0.966·29-s − 1.11i·31-s + 0.656·35-s + 1.40i·37-s − 1.41i·41-s + 0.149·43-s − 0.951i·47-s − 1.15·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556871059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556871059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.54 + 0.660i)T \) |
| good | 7 | \( 1 + 3.88iT - 7T^{2} \) |
| 11 | \( 1 - 1.54iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.86iT - 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 - 8.56iT - 37T^{2} \) |
| 41 | \( 1 + 9.08iT - 41T^{2} \) |
| 43 | \( 1 - 0.980T + 43T^{2} \) |
| 47 | \( 1 + 6.52iT - 47T^{2} \) |
| 53 | \( 1 + 6.44T + 53T^{2} \) |
| 59 | \( 1 - 4.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 6.97iT - 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.43iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 8.56iT - 83T^{2} \) |
| 89 | \( 1 + 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−8.709125234480929070955319704239, −7.973856469274578264471913558754, −7.22857823775016141467566672477, −6.62482424578790494525679453021, −5.82318756662032791539946463017, −4.60323282406559557783553497683, −3.98554237600106455541338146060, −3.12920944726107936351752383177, −1.84166882085968093845363408931, −0.56517850377417058969483218714,
1.29353098398290518875738647531, 2.37715235556412357354197549621, 3.38539307946793023244940068001, 4.36266850789710813432278711129, 5.40770103827613335041664039804, 5.94434041793353008687454180086, 6.61051533141047255617364681067, 7.961636583332536010065412608558, 8.419933562945738606728392776175, 9.024323733099214977172254880954
