| L(s) = 1 | + 3-s − 5-s + 0.845·7-s + 9-s + 4.55·11-s + 5.23·13-s − 15-s + 4.72·17-s + 5.54·19-s + 0.845·21-s − 23-s + 25-s + 27-s + 7.06·29-s + 1.83·31-s + 4.55·33-s − 0.845·35-s − 9.93·37-s + 5.23·39-s + 5.71·41-s − 4.78·43-s − 45-s − 5.54·47-s − 6.28·49-s + 4.72·51-s − 14.4·53-s − 4.55·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.319·7-s + 0.333·9-s + 1.37·11-s + 1.45·13-s − 0.258·15-s + 1.14·17-s + 1.27·19-s + 0.184·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.31·29-s + 0.329·31-s + 0.793·33-s − 0.142·35-s − 1.63·37-s + 0.838·39-s + 0.892·41-s − 0.729·43-s − 0.149·45-s − 0.808·47-s − 0.897·49-s + 0.661·51-s − 1.99·53-s − 0.614·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.195150587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.195150587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 0.845T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 + 5.06T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 - 7.85T + 67T^{2} \) |
| 71 | \( 1 + 3.71T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 + 2.33T + 97T^{2} \) |
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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
−8.161956183691914813552589023361, −7.63478071389060052159225067790, −6.68750251649406870117060206249, −6.19473447067083596135562275971, −5.16112458275634437687960224084, −4.37567095419334721432429136212, −3.38433344288117262657466253226, −3.27996776677865521372040505375, −1.61711253381787129786102791470, −1.07680411317122498738490689733,
1.07680411317122498738490689733, 1.61711253381787129786102791470, 3.27996776677865521372040505375, 3.38433344288117262657466253226, 4.37567095419334721432429136212, 5.16112458275634437687960224084, 6.19473447067083596135562275971, 6.68750251649406870117060206249, 7.63478071389060052159225067790, 8.161956183691914813552589023361
