| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 7-s − 2.23·8-s + 9-s + 2.80·11-s − 0.618·12-s − 13-s − 1.61·14-s − 4.85·16-s + 2.01·17-s + 1.61·18-s − 2.34·19-s + 21-s + 4.53·22-s + 0.939·23-s + 2.23·24-s − 1.61·26-s − 27-s − 0.618·28-s − 5.69·29-s + 0.895·31-s − 3.38·32-s − 2.80·33-s + 3.25·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.845·11-s − 0.178·12-s − 0.277·13-s − 0.432·14-s − 1.21·16-s + 0.487·17-s + 0.381·18-s − 0.537·19-s + 0.218·21-s + 0.967·22-s + 0.195·23-s + 0.456·24-s − 0.317·26-s − 0.192·27-s − 0.116·28-s − 1.05·29-s + 0.160·31-s − 0.597·32-s − 0.488·33-s + 0.557·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.234941297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.234941297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 11 | \( 1 - 2.80T + 11T^{2} \) |
| 17 | \( 1 - 2.01T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 0.939T + 23T^{2} \) |
| 29 | \( 1 + 5.69T + 29T^{2} \) |
| 31 | \( 1 - 0.895T + 31T^{2} \) |
| 37 | \( 1 - 5.52T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 2.77T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 0.821T + 89T^{2} \) |
| 97 | \( 1 - 1.98T + 97T^{2} \) |
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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
−7.75592753826545474883991951403, −6.97581276818259527798973564740, −6.30340727247667022965224458339, −5.83039235589245102228109656520, −5.10909401904738429748573112214, −4.36814024240719053193099255260, −3.79815998776423379135720769160, −3.03118431940881103007065324075, −1.98854654944513581473999907990, −0.65232794399704564605918304597,
0.65232794399704564605918304597, 1.98854654944513581473999907990, 3.03118431940881103007065324075, 3.79815998776423379135720769160, 4.36814024240719053193099255260, 5.10909401904738429748573112214, 5.83039235589245102228109656520, 6.30340727247667022965224458339, 6.97581276818259527798973564740, 7.75592753826545474883991951403
