| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 5.80·11-s − 5.05·13-s − 15-s + 0.755·17-s − 1.80·19-s − 21-s − 8.85·23-s + 25-s + 27-s − 0.755·29-s + 10.6·31-s + 5.80·33-s + 35-s + 0.755·37-s − 5.05·39-s − 9.61·41-s + 1.24·43-s − 45-s + 8.85·47-s + 49-s + 0.755·51-s + 7.80·53-s − 5.80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 1.75·11-s − 1.40·13-s − 0.258·15-s + 0.183·17-s − 0.414·19-s − 0.218·21-s − 1.84·23-s + 0.200·25-s + 0.192·27-s − 0.140·29-s + 1.91·31-s + 1.01·33-s + 0.169·35-s + 0.124·37-s − 0.808·39-s − 1.50·41-s + 0.189·43-s − 0.149·45-s + 1.29·47-s + 0.142·49-s + 0.105·51-s + 1.07·53-s − 0.782·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.109657112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109657112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 - 0.755T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 0.755T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 - 1.24T + 43T^{2} \) |
| 47 | \( 1 - 8.85T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.05T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7.61T + 83T^{2} \) |
| 89 | \( 1 - 8.10T + 89T^{2} \) |
| 97 | \( 1 - 5.05T + 97T^{2} \) |
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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
−8.096122655267081442445044450972, −7.21137814226256411063663491084, −6.72791566501989740472061382898, −6.03965720609195693779251692120, −4.99700768428755630499183739305, −4.06524320990452699721949747629, −3.82917006401162366868248400190, −2.69919829450832994211620316902, −1.96148625244070056526099949183, −0.71869627202001076490827564458,
0.71869627202001076490827564458, 1.96148625244070056526099949183, 2.69919829450832994211620316902, 3.82917006401162366868248400190, 4.06524320990452699721949747629, 4.99700768428755630499183739305, 6.03965720609195693779251692120, 6.72791566501989740472061382898, 7.21137814226256411063663491084, 8.096122655267081442445044450972
