| L(s) = 1 | − 3-s − 3.90·5-s + 0.0794·7-s + 9-s − 1.50·11-s + 3.90·15-s − 6.52·17-s + 0.786·19-s − 0.0794·21-s − 7.03·23-s + 10.2·25-s − 27-s + 6.15·29-s − 1.43·31-s + 1.50·33-s − 0.310·35-s − 9.23·37-s − 7.75·41-s + 1.06·43-s − 3.90·45-s − 12.7·47-s − 6.99·49-s + 6.52·51-s + 2.73·53-s + 5.87·55-s − 0.786·57-s − 10.3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.74·5-s + 0.0300·7-s + 0.333·9-s − 0.453·11-s + 1.00·15-s − 1.58·17-s + 0.180·19-s − 0.0173·21-s − 1.46·23-s + 2.05·25-s − 0.192·27-s + 1.14·29-s − 0.258·31-s + 0.261·33-s − 0.0524·35-s − 1.51·37-s − 1.21·41-s + 0.162·43-s − 0.582·45-s − 1.86·47-s − 0.999·49-s + 0.913·51-s + 0.375·53-s + 0.792·55-s − 0.104·57-s − 1.35·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3585770297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3585770297\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 - 0.0794T + 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.786T + 19T^{2} \) |
| 23 | \( 1 + 7.03T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 1.06T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 4.16T + 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 - 4.75T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.213786094176718627294322056249, −7.87052375857692621089668709114, −6.80850475057607143350069741157, −6.55919319607380232725193469912, −5.23994208737904123015801505295, −4.64674314239747487298797171113, −3.94027966150774706194639887340, −3.17204999985260139209035647722, −1.89656010645469686629751257495, −0.33833339390183669594404512242,
0.33833339390183669594404512242, 1.89656010645469686629751257495, 3.17204999985260139209035647722, 3.94027966150774706194639887340, 4.64674314239747487298797171113, 5.23994208737904123015801505295, 6.55919319607380232725193469912, 6.80850475057607143350069741157, 7.87052375857692621089668709114, 8.213786094176718627294322056249
