| L(s) = 1 | − 2-s + 2.44·3-s + 4-s − 2.44·6-s − 4.44·7-s − 8-s + 2.99·9-s + 4.89·11-s + 2.44·12-s − 4·13-s + 4.44·14-s + 16-s + 4.89·17-s − 2.99·18-s + 3.55·19-s − 10.8·21-s − 4.89·22-s + 8.89·23-s − 2.44·24-s + 4·26-s − 4.44·28-s − 1.55·31-s − 32-s + 11.9·33-s − 4.89·34-s + 2.99·36-s − 37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.999·6-s − 1.68·7-s − 0.353·8-s + 0.999·9-s + 1.47·11-s + 0.707·12-s − 1.10·13-s + 1.18·14-s + 0.250·16-s + 1.18·17-s − 0.707·18-s + 0.814·19-s − 2.37·21-s − 1.04·22-s + 1.85·23-s − 0.499·24-s + 0.784·26-s − 0.840·28-s − 0.278·31-s − 0.176·32-s + 2.08·33-s − 0.840·34-s + 0.499·36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.875186236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875186236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 - 8.89T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 6.44T + 79T^{2} \) |
| 83 | \( 1 - 9.55T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.242671272269700558600008802731, −8.779368417076214268273552862823, −7.66807182309817955126179829280, −7.08697567606439108030056513032, −6.46955316109337030641147437160, −5.25563231020957956959907862060, −3.67460659006294062179036263406, −3.27560099314822594937463187889, −2.41158368906432140772939204588, −0.991227318871040205129127355935,
0.991227318871040205129127355935, 2.41158368906432140772939204588, 3.27560099314822594937463187889, 3.67460659006294062179036263406, 5.25563231020957956959907862060, 6.46955316109337030641147437160, 7.08697567606439108030056513032, 7.66807182309817955126179829280, 8.779368417076214268273552862823, 9.242671272269700558600008802731
