| L(s) = 1 | + 1.08·2-s − 3-s − 0.812·4-s − 1.08·6-s − 3.08·7-s − 3.06·8-s + 9-s − 1.14·11-s + 0.812·12-s + 4.07·13-s − 3.36·14-s − 1.71·16-s + 4.62·17-s + 1.08·18-s − 5.96·19-s + 3.08·21-s − 1.25·22-s − 2.32·23-s + 3.06·24-s + 4.44·26-s − 27-s + 2.50·28-s − 5.28·29-s − 0.589·31-s + 4.26·32-s + 1.14·33-s + 5.04·34-s + ⋯ |
| L(s) = 1 | + 0.770·2-s − 0.577·3-s − 0.406·4-s − 0.444·6-s − 1.16·7-s − 1.08·8-s + 0.333·9-s − 0.346·11-s + 0.234·12-s + 1.13·13-s − 0.899·14-s − 0.428·16-s + 1.12·17-s + 0.256·18-s − 1.36·19-s + 0.673·21-s − 0.266·22-s − 0.484·23-s + 0.625·24-s + 0.871·26-s − 0.192·27-s + 0.473·28-s − 0.981·29-s − 0.105·31-s + 0.753·32-s + 0.199·33-s + 0.864·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.264374708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264374708\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 0.589T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 9.49T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 - 3.18T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 - 4.52T + 97T^{2} \) |
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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
−9.379396784995201771520818622655, −8.487815171617198380709620830653, −7.58629966677895544089480812681, −6.33318486985220735055372076378, −6.06024963017629916046414394513, −5.30140094988542649831231437884, −4.11047631084457508527443908011, −3.69090689988160441000742583989, −2.54211683051286553015614944683, −0.68504878245081997097536531131,
0.68504878245081997097536531131, 2.54211683051286553015614944683, 3.69090689988160441000742583989, 4.11047631084457508527443908011, 5.30140094988542649831231437884, 6.06024963017629916046414394513, 6.33318486985220735055372076378, 7.58629966677895544089480812681, 8.487815171617198380709620830653, 9.379396784995201771520818622655
