| L(s) = 1 | − 2-s − i·3-s + 4-s + i·6-s + 4.60·7-s − 8-s − 9-s − i·12-s + 3.60i·13-s − 4.60·14-s + 16-s − 4.60i·17-s + 18-s − 4.60i·19-s − 4.60i·21-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.74·7-s − 0.353·8-s − 0.333·9-s − 0.288i·12-s + 0.999i·13-s − 1.23·14-s + 0.250·16-s − 1.11i·17-s + 0.235·18-s − 1.05i·19-s − 1.00i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496625656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496625656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 3.60iT \) |
| good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 4.60iT - 17T^{2} \) |
| 19 | \( 1 + 4.60iT - 19T^{2} \) |
| 23 | \( 1 + 1.39iT - 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 + 3.21iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 9.21iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 + 9.21iT - 71T^{2} \) |
| 73 | \( 1 + 1.39T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
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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
−9.167756159299580673465285042047, −8.098586508991576797281032382099, −7.64455808656348861937575987534, −6.96842236451774125825538115105, −6.01442907924315208823330848514, −4.98771780387529298047014204053, −4.27074843773739220350446643137, −2.62520140197357839201612056485, −1.89558761345637549971403176455, −0.789476706413939049664863182814,
1.20199719115249094075275423368, 2.17756307531788737357164183128, 3.50643149976493371832692449099, 4.41610914226553336469663320287, 5.42534779433789194346228609953, 5.94858988028833324703397323805, 7.33755643807672661832003843304, 7.997377040619744982778380477074, 8.430379890539206441405590076027, 9.228098031568900277033439917289
